Orthogonal time frequency space modulation system

ABSTRACT

A system and method for orthogonal time frequency space communication and waveform generation. The method includes receiving a plurality of information symbols and creating a plurality of modulation symbols by using each of the plurality information symbols to modulate one of a plurality of two-dimensional basis functions on a time-frequency plane Each of the plurality of two-dimensional basis functions is uniquely associated with one of the plurality of information symbols. The method further includes generating a transmit waveform comprised of a plurality of pulse waveforms. Each of the plurality of pulse waveforms corresponds to a combination of one of the plurality of modulation symbols and one of a plurality of time-translated and frequency-modulated versions of a fundamental transmit pulse.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application claims the benefit of priority under 35 U.S.C. §119(e) of U.S. Provisional Application No. 62/159,853, entitledORTHOGONAL TIME FREQUENCY SPACE OTFS MODULATION, filed May 11, 2015, ofU.S. Provisional Application No. 62/160,257, entitled SYSTEMS ANDMETHODS FOR SYMPLECTIC ORTHOGONAL TIME FREQUENCY SHIFTING MODULATION ANDTRANSMISSION OF DATA, filed May 12, 2015, of U.S. ProvisionalApplication No. 62/173,801, entitled SYSTEMS AND METHODS FOR SYMPLECTICORTHOGONAL TIME FREQUENCY SHIFTING MODULATION AND TRANSMISSION OF DATA,filed Jun. 10, 2015, of U.S. Provisional Application No. 62/182,372,entitled OTFS A NEW MODULATION FOR 5G, filed Jun. 19, 2015, and of U.S.Provisional Application No. 62/215,663, entitled ORTHOGONAL TIMEFREQUENCY SPACE COMMUNICATION SYSTEM AND METHOD, filed Sep. 8, 2015, thecontents of each of which are hereby incorporated by reference in theirentirety for all purposes. The present application is acontinuation-in-part of U.S. application Ser. No. 14/709,377, entitledMODULATION AND EQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTINGCOMMUNICATIONS SYSTEM, filed May 11, 2015, which is a continuation ofU.S. patent application Ser. No. 13/927,086, entitled MODULATION ANDEQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONSSYSTEM, filed on Jun. 25, 2013, which claims the benefit of priorityunder 35 U.S.C. § 119(e) of U.S. Provisional Application Ser. No.61/664,020, entitled MODULATION AND EQUALIZATION IN AN ORTHONORMALTIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM, filed Jun. 25, 2012, ofU.S. Provisional Application Ser. No. 61/801,398, entitled MODULATIONAND EQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTINGCOMMUNICATIONS SYSTEM, filed Mar. 15, 2013, of U.S. ProvisionalApplication Ser. No. 61/801,366, entitled MODULATION AND EQUALIZATION INAN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM, filed Mar.15, 2013, of U.S. Provisional Application Ser. No. 61/801,435, entitledMODULATION AND EQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTINGCOMMUNICATIONS SYSTEM, filed Mar. 15, 2013, of U.S. ProvisionalApplication Ser. No. 61/801,495, entitled MODULATION AND EQUALIZATION INAN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM, filed Mar.15, 2013, of U.S. Provisional Application Ser. No. 61/801,994, entitledMODULATION AND EQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTINGCOMMUNICATIONS SYSTEM, filed Mar. 15, 2013, and of U.S. ProvisionalApplication Ser. No. 61/801,968, entitled MODULATION AND EQUALIZATION INAN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM, filed Mar.15, 2013, the contents of each of which are hereby incorporated byreference in their entirety for all purposes.

FIELD

This disclosure generally relates to communications protocols andmethods, and more particularly relates to methods for modulation andrelated processing of signals used for wireless and other forms ofcommunication.

BACKGROUND

Fourth generation (4G) wireless networks have served the public well,providing ubiquitous access to the Internet and enabling the explosionof mobile apps, smartphones and sophisticated data intensiveapplications like mobile video. This continues the evolution of cellulartechnologies, where each new generation brings substantial benefits tothe public, enabling significant gains in productivity, convenience, andquality of life.

Looking ahead to the demands that the ever increasing and diverse datausage is placing on existing networks, it is becoming clear to theindustry that current 4G networks will not be able to support theforeseen needs in data usage. This is in part because data trafficvolume has been, and continues to, increase at an exponential rate.Moreover, new applications such as, for example, immersive reality andremote robotic operation, coupled with the ongoing expansion of mobilevideo, are expected to overwhelm the carrying capacity of currentnetwork systems. One of the goals of 5G system design is to be able toeconomically scale the capabilities of networks in dense urban settings(e.g., to 750 Gbps per sq. Km), which is not possible using technologywhich has been commercially deployed.

In addition to being able to handle larger volumes of data, nextgeneration systems will need to improve the quality of data delivery inorder to support desired future applications. The public is increasinglycoming to expect that wireless networks provide a near “wireline”experience to the untethered user. This may translate to, for example, arequirement of 50+ Mbps throughout coverage areas (i.e., even at celledges), which will require advanced interference mitigation technologiesto be realized.

Another aspect of the quality of user experience is mobility. Thethroughput of current wireless networks tends to be dramatically reducedin tandem with increased mobile speeds due to Doppler effects. Future 5Gsystems aim to not only increase supported speeds up to 500 Km/h forhigh speed trains and aviation, but to also support a host of newautomotive applications for vehicle-to-vehicle andvehicle-to-infrastructure communications.

While the support of increased and higher quality data traffic isnecessary for wireless networks to continue supporting user needs,carriers are also exploring new applications that will enable newrevenues and innovative use cases. These include the automotive andsmart infrastructure applications discussed above. Other desiredapplications include the deployment of public safety ultra-reliablenetworks, the use of cellular networks to support the sunset of thePSTN, and the like. Moreover, it is anticipated the 5G networks willusher in the deployment of large numbers of Internet connected devices,also known as the Internet of Things (IoT). However, existing networksare not designed to support a very large number of connected deviceswith very low traffic per device.

SUMMARY

In one aspect the disclosure relates to a method of generating awaveform for transmission over a communication channel. The methodincludes receiving a plurality of information symbols and creating aplurality of modulation symbols by using each of the pluralityinformation symbols to modulate one of a plurality of two-dimensionalbasis functions on a time-frequency plane Each of the plurality oftwo-dimensional basis functions is uniquely associated with one of theplurality of information symbols. The method further includes generatinga transmit waveform comprised of a plurality of pulse waveforms, each ofthe plurality of pulse waveforms corresponding to a combination of oneof the plurality of modulation symbols and one of a plurality oftime-translated and frequency-modulated versions of a fundamentaltransmit pulse.

In one implementation each of the plurality of time-translated andfrequency-modulated versions of the fundamental transmit pulse isassociated with a time translation by one of N multiples of T and afrequency modulation by one of M multiples of Δƒ, where the transmitwaveform is of a total duration of NT seconds and a total bandwidth ofMΔƒ Hz. The fundamental transmit pulse may have the property of beingorthogonal to translations by a multiple of time T and modulation by amultiple of Δƒ.

In another aspect the disclosure relates to a communication deviceincluding a wireless transmitter, a processor and a memory includingprogram code executable by the processor. The program code includes codefor causing the processor to receive a plurality of information symbols.The program code further includes code for causing the processor tocreate a plurality of modulation symbols by using each of the pluralityinformation symbols to modulate one of a plurality of two-dimensionalbasis functions on a time-frequency plane. Each of the plurality oftwo-dimensional basis functions is uniquely associated with one of theplurality of information symbols. The program code further causes theprocessor to generate a transmit waveform comprised of a plurality ofpulse waveforms, each of the plurality of pulse waveforms correspondingto a combination of one of the plurality of modulation symbols and oneof a plurality of time-translated and frequency-modulated versions of afundamental transmit pulse. The program code further includes code forcausing the processor to provide the transmit waveform to the wirelesstransmitter.

The disclosure is also directed to a method for receiving, at acommunication receiver, one or more modulated waveforms. The methodfurther includes performing matched filtering of samples of the one ormore modulated waveforms with respect to a receive pulse to produceestimated time-frequency modulation symbols. Each of the estimatedtime-frequency modulation symbols corresponds to modulation of one of aplurality of orthogonal two-dimensional basis functions by one of aplurality of information symbols. The method further includes projectingthe estimates of the time-frequency modulation symbols on the pluralityof orthogonal two-dimensional basis functions in order to obtainestimates of the plurality of information symbols.

In one implementation the method may further include performingwindowing and periodization operations with respect to the estimatedtime-frequency modulation symbols. In addition, the projecting operationmay include performing a symplectic Fourier transform operation withrespect to a periodic sequence comprised of the estimates of thetime-frequency modulation symbols.

In yet another aspect the disclosure pertains to a communication deviceincluding a wireless receiver configured to receive one or moremodulated waveforms, a processor and a memory including program codeexecutable by the processor. The program code includes code for causingthe processor to receive, from the wireless receiver, samples of one ormore modulated waveforms. The code further includes code for causing theprocessor to matched filter samples of the one or more modulatedwaveforms with respect to a receive pulse to produce estimatedtime-frequency modulation symbols. Each of the estimated time-frequencymodulation symbols corresponds to modulation of one of a plurality oforthogonal two-dimensional basis functions by one of a plurality ofinformation symbols. The program code further includes code for causingthe processor to project the estimated time-frequency modulation symbolson the plurality of orthogonal two-dimensional basis functions in orderto obtain estimates of the plurality of information symbols.

In one implementation the program code may further include code forperforming windowing and periodization operations with respect to theestimated time-frequency modulation symbols. In addition, the code mayinclude code for causing the processor to perform a symplectic Fouriertransform operation with respect to a periodic sequence comprised of theestimated time-frequency modulation symbols.

BRIEF DESCRIPTION OF THE DRAWINGS

For a better understanding of the nature and objects of variousembodiments of the invention, reference should be made to the followingdetailed description taken in conjunction with the accompanyingdrawings, wherein:

FIG. 1A illustrates an example of a wireless communication system thatmay exhibit time/frequency selective fading.

FIG. 1B provides a high-level representation of a conventionaltransceiver which could be utilized in the wireless communication systemof FIG. 1A.

FIG. 2A shows the time-varying impulse response for an acceleratingreflector in a channel represented by a one-dimensional channel model ina (τ, t) coordinate system.

FIG. 2B shows the same channel represented using a time invariantimpulse response in a delay-Doppler (τ, v) coordinate system.

FIG. 3 is a block diagram of components of an exemplary OTFScommunication system.

FIG. 4 represents a conceptual implementation of a Heisenberg transformin an OTFS transmitter and a Wigner transform in an OTFS receiver.

FIG. 5 illustratively represents an exemplary embodiment of OTFSmodulation, including the transformation of the time-frequency plane tothe Doppler-delay plane.

FIG. 6 shows a discrete impulse in the OTFS domain which is used forpurposes of channel estimation.

FIG. 7 illustrates two different basis functions belonging to differentusers, each of which spans the entire time-frequency frame.

FIGS. 8 and 9 illustrate multiplexing multiple users in thetime-frequency domain by allocating different resource blocks orsubframes to different users in an interleaved manner.

FIG. 10 illustrates components of an exemplary OTFS transceiver.

FIG. 11 illustrates a comparison of bit error rates (BER) predicted by asimulation of a TDMA system and an OTFS system.

FIG. 12 is a flowchart representative of the operations performed by anexemplary OTFS transceiver.

FIG. 13 illustrates functioning of an OTFS modulator as an orthogonalmap disposed to transform a two-dimensional time-frequency matrix into atransmitted waveform.

FIG. 14 illustrates operation of an OTFS demodulator in transforming areceived waveform into a two-dimensional time-frequency matrix inaccordance with an orthogonal map.

FIG. 15 illustratively represents a pulse train included within a pulsewaveform produced by an OTFS modulator.

FIG. 16 depicts a two-dimensional decision feedback equalizer configuredto perform a least means square (LMS) equalization procedure.

FIGS. 17A-17D depict an OTFS transmitter and receiver and the operationof each with respect to associated time-frequency grids.

FIG. 18 includes a set of bar diagrams representing a two-dimensionalimpulse response realizing a finite modulation equivalent channel, atransmitted information vector x and a received information vector y.

FIG. 19 illustrates transmission of a 2D Fourier transformed informationmanifold represented by an N×M structure over M frequency bands during Ntime periods of duration Tμ.

FIG. 20 shows an example of M filtered OTFS frequency bands beingsimultaneously transmitted according to various smaller time slices Tμ.

FIG. 21 provides an additional example of OTFS waveforms beingtransmitted according to various smaller time slices Tμ.

FIG. 22 provides a block diagrammatic representation of an exemplaryprocess of OTFS transmission and reception.

FIG. 23 illustrates represents an exemplary structure of finite OTFSmodulation map.

FIGS. 24A and 24B respectively illustrate a standard communicationlattice and the reciprocal of the standard communication lattice.

FIG. 25 illustratively represents a standard communication torus.

FIG. 26 illustratively represents a standard communication finite torus.

FIG. 27 illustrates an exemplary structure of an OTFS modulation map.

FIG. 28 illustrates a frequency domain interpretation of an OTFSmodulation block.

DETAILED DESCRIPTION

As is discussed below, embodiments of orthogonal time frequency space(OTFS) modulation involve transmitting each information symbol bymodulating a two-dimensional (2D) basis function on the time-frequencyplane. In exemplary embodiments the modulation basis function set isspecifically derived to best represent the dynamics of the time varyingmultipath channel. In this way OTFS transforms the time-varyingmultipath channel into a time invariant delay-Doppler two dimensionalconvolution channel. This effectively eliminates the difficulties intracking time-varying fading in, for example, communications involvinghigh speed vehicles.

OTFS increases the coherence time of the channel by orders of magnitude.It simplifies signaling over the channel using well studied AWGN codesover the average channel SNR. More importantly, it enables linearscaling of throughput with the number of antennas in moving vehicleapplications due to the inherently accurate and efficient estimation ofchannel state information (CSI). In addition, since the delay-Dopplerchannel representation is very compact, OTFS enables massive MIMO andbeamforming with CSI at the transmitter for four, eight, and moreantennas in moving vehicle applications. The CSI information needed inOTFS is a fraction of what is needed to track a time varying channel.

As will be appreciated from the discussion below, one characteristic ofOTFS is that a single QAM symbol may be spread over multiple time and/orfrequency points. This is a key technique to increase processing gainand in building penetration capabilities for IoT deployment and PSTNreplacement applications. Spreading in the OTFS domain allows spreadingover wider bandwidth and time durations while maintaining a stationarychannel that does not need to be tracked over time.

These benefits of OTFS will become apparent once the basic conceptsbehind OTFS are understood. There is a rich mathematical foundation ofOTFS that leads to several variations; for example it can be combinedwith OFDM or with multicarrier filter banks. Prior to proceeding to adetailed discussion of OTFS, various drawbacks of communication systemspredicated on one-dimensional channel models are first described.

FIG. 1A illustrates an example of a wireless communication system 100that may exhibit time/frequency selective fading. The system 100includes a transmitter 110 (e.g., a cell phone tower) and a receiver 120(e.g., a cell phone). The scenario illustrated in FIG. 1 includesmultiple pathways (multi-path) that the signal transmitted from thetransmitter 100 travels through before arriving at the receiver 100. Afirst pathway 130 reflects through a tree 132, second pathway 140reflects off of a building 142 and a third pathway 150 reflects off of asecond building 152. A fourth pathway 160 reflects off of a moving car162. Because each of the pathways 130, 140, 150 and 160 travels adifferent distance, and is attenuated or faded at a different level andat a different frequency, when conventionally configured the receiver120 may drop a call or at least suffer low throughput due to destructiveinterference of the multi-path signals.

Turning now to FIG. 1B, a high-level representation is provided of aconventional transceiver 200 which could be utilized in the wirelesscommunication system 100 of FIG. 1A. The transceiver 200 could, forexample, operate in accordance with established protocols fortime-division multiple access (TDMA), code-division multiple access(CDMA) or orthogonal frequency-division multiple access (OFDM) systems.In conventional wireless communication systems such as TDMA, CDMA, andOFDM) systems, the multipath communication channel 210 between atransmitter 204 and a receiver 208 is represented by a one-dimensionalmodel. In these systems channel distortion is characterized using aone-dimensional representation of the impulse response of thecommunication channel. The transceiver 200 may include a one-dimensionalequalizer 220 configured to at least partially remove this estimatedchannel distortion from the one-dimensional output data stream 230produced by the receiver 208.

Unfortunately, use of a one-dimensional channel model presents a numberof fundamental problems. First, the one-dimensional channel modelsemployed in existing communication systems are non-stationary; that is,the symbol-distorting influence of the communication channel changesfrom symbol to symbol. In addition, when a channel is modeled in onlyone dimension it is likely and possible that certain received symbolswill be significantly lower in energy than others due to “channelfading”. Finally, one-dimensional channel state information (CSI)appears random and much of it is estimated by interpolating betweenchannel measurements taken at specific points, thus rendering theinformation inherently inaccurate. These problems are only exacerbatedin multi-antenna (MIMO) communication systems. As is discussed below,embodiments of the OTFS method described herein can be used tosubstantially overcome the fundamental problems arising from use of aone-dimensional channel model.

The multipath fading channel is commonly modeled one-dimensionally inthe baseband as a convolution channel with a time varying impulseresponser(t)=∫

(τ,t)s(t−τ)dτ  (1)where s(t) and r(t) represent the complex baseband channel input andoutput respectively and where

(τ, t) is the complex baseband time varying channel response.

This representation, while general, does not give us insight into thebehavior and variations of the time varying impulse response. A moreuseful and insightful model, which is also commonly used for Dopplermultipath doubly fading channels isr(t)=∫∫h(τ,v)e ^(j2τv(t−1τ)) s(t−τ)dvdτ  (2)

In this representation, the received signal is a superposition ofreflected copies of the transmitted signal, where each copy is delayedby the path delay τ, frequency shifted by the Doppler shift v andweighted by the time-independent delay-Doppler impulse response h(τ, v)for that τ and v. In addition to the intuitive nature of thisrepresentation, Eq. (2) maintains the generality of Eq. (1). In otherwords it can represent complex Doppler trajectories, like acceleratingvehicles, reflectors etc. This can be seen if we express the timevarying impulse response as a Fourier expansion with respect to the timevariable t

(τ,t)=∫h(τ,v)e ^(j2πvt) dt  (2)

Substituting (3) in (1) we obtain Eq. (2) after some manipulation. As anexample, FIG. 2A shows the time-varying impulse response for anaccelerating reflector in the (τ, t) coordinate system, while FIG. 2Bshows the same channel represented as a time invariant impulse responsein the (τ, v) coordinate system.

An important feature revealed by these two figures is how compact the(τ, v) representation is compared to the (τ, t) representation. This hasimportant implications for channel estimation, equalization and trackingas will be discussed later.

Notice that while h(τ, v) is, in fact, time-independent, the operationon s(t) is still time varying, as can be seen by the effect of theexplicit complex exponential function of time in Eq. (2). Inimplementation the disclosed modulation scheme contemplates anappropriate choice of orthogonal basis functions that render the effectsof this channel to become truly time-independent in the domain definedby those basis functions. The proposed scheme has the following highlevel outline.

First, let us consider a set of orthonormal basis functions ϕ_(τ,v)(t)indexed by τ, v which are orthogonal to translation and modulation,i.e.,ϕ_(τ,v)(t−τ ₀)=ϕ_(τ+τ) ₀ _(,v)(t)e ^(j2πv) ⁰ ^(t)ϕ_(τ,v)(t)=ϕ_(τ,v−v) ₀ (t)  (4)and let us consider the transmitted signal as a superposition of thesebasis functionss(t)=∫∫x(τ,v)ϕ_(τ,v)(t)dτdv  (5)where the weights x(τ, v) represent the information bearing signal to betransmitted. After the transmitted signal of (5) goes through the timevarying channel of Eq. (2) we obtain a superposition of delayed andmodulated versions of the basis functions, which due to (4) results in

$\begin{matrix}\begin{matrix}{{r(t)} = {\int{\int{{h\left( {\tau,v} \right)}e^{j\; 2\pi\;{v{({t - \tau})}}}{s\left( {t - \tau} \right)}{dvd}\;\tau}}}} \\{= {\int{\int{{\phi_{\tau,v}(t)}\left\{ {{h\left( {\tau,v} \right)}*{x\left( {\tau,v} \right)}} \right\} d\;\tau\;{dv}}}}}\end{matrix} & (6)\end{matrix}$where * denotes two dimensional convolution. Eq. (6) can be thought ofas a generalization of the convolution relationship for linear timeinvariant systems, using one dimensional exponentials as basisfunctions. Notice that the term in brackets can be recovered at thereceiver by matched filtering against each basis function ϕ_(τ,v)(t). Inthis way a two dimensional channel relationship is established in the(τ, v) domainy(τ,v)=h(τ,v)*x(τ,v)  (7)where y(τ, v) is the receiver two dimensional matched filter output.Notice also, that in this domain the channel is described by a timeinvariant two-dimensional convolution.

A final different interpretation of the wireless channel will also beuseful in what follows. Let us consider s(t) and r(t) as elements of theHilbert space of square integrable functions

. Then Eq. (2) can be interpreted as a linear operator on

acting on the input s(t), parametrized by the impulse response h(τ, v),and producing the output r(t):

Notice that although the operator is linear, it is not time-invariant.If there is no Doppler, i.e., if h(v, τ)=h(0, τ)δ(v), then Eq. (2)reduces to a time invariant convolution. Also notice that while for timeinvariant systems the impulse response is parameterized by onedimension, in the time varying case we have a two dimensional impulseresponse. While in the time invariant case the convolution operatorproduces a superposition of delays of the input s(t), (hence theparameterization is along the one dimensional delay axis) in the timevarying case we have a superposition of delay-and-modulate operations asseen in Eq. (2) (hence the parameterization is along the two dimensionaldelay and Doppler axes). This is a major difference which makes the timevarying representation non-commutative (in contrast to the convolutionoperation which is commutative), and complicates the treatment of timevarying systems.

One important point of Eq. (8) is that the operator Π_(h)(⋅) can becompactly parametrized by a two dimensional function h(τ, v), providingan efficient, time-independent description of the channel. Typicalchannel delay spreads and Doppler spreads are a very small fraction ofthe symbol duration and subcarrier spacing of multicarrier systems.

The representation of time varying systems defined by equations (2) and(8) may be characterized as a Heisenberg representation. In this regardit may be shown that every linear operator (eq. (8)) can beparameterized by some impulse response as in equation (2).

OTFS Modulation Over the Doppler Multipath Channel

The time variation of the channel introduces significant difficulties inwireless communications related to channel acquisition, tracking,equalization and transmission of channel state information (CSI) to thetransmit side for beamforming and MIMO processing. We herein develop amodulation domain based on a set of orthonormal basis functions overwhich we can transmit the information symbols, and over which theinformation symbols experience a static, time invariant, two dimensionalchannel for the duration of the packet or burst transmission. In thatmodulation domain, the channel coherence time is increased by orders ofmagnitude and the issues associated with channel fading in the time orfrequency domain in SISO or MIMO systems are significantly reduced.

FIG. 3 is a block diagram of components of an exemplary OTFScommunication system 300. As shown, the system 300 includes atransmitter 310 and a receiver 330. The transmitting device 310 and thereceiving device 330 include first and second OTFS transceivers 315-1and 315-2, respectively. The OTFS transceivers 315-1 and 315-2communicate, either unidirectionally or bidirectionally, viacommunication channel 320 in the manner described herein. Although inthe exemplary embodiments described herein the system 300 may comprise awireless communication system, in other embodiments the communicationchannel may comprise a wired communication channel such as, for example,a communication channel within a fiber optic or coaxial cable. As wasdescribed above, the communication channel 320 may include multiplepathways and be characterized by time/frequency selective fading.

The components of the OTFS transceiver may be implemented in hardware,software, or a combination thereof. For a hardware implementation, theprocessing units may be implemented within one or more applicationspecific integrated circuits (ASICs), digital signal processors (DSPs),digital signal processing devices (DSPDs), programmable logic devices(PLDs), field programmable gate arrays (FPGAs), processors, controllers,micro-controllers, microprocessors, other electronic units designed toperform the functions described above, and/or a combination thereof.

Referring now to FIG. 3B, there is provided a pictorial view of the twotransformations that constitute an exemplary form of OTFS modulation. Itshows at a high level the signal processing steps that are required at atransmitter, such as the transmitter 310, and a receiver, such as thereceiver 330. It also includes the parameters that define each step,which will become apparent as we further expose each step. Further, FIG.3C shows a block diagram of the different processing stages at thetransmitter and receiver and establishes the notation that will be usedfor the various signals.

We initially describe the transform which relates the waveform domain tothe time-frequency domain.

The Heisenberg Transform

Our purpose in this section is to construct an appropriate transmitwaveform which carries information provided by symbols on a grid in thetime-frequency plane. Our intent in developing this modulation scheme isto transform the channel operation to an equivalent operation on thetime-frequency domain with two important properties: (i) the channel isorthogonalized on the time-frequency grid; and (ii) the channel timevariation is simplified on the time-frequency grid and can be addressedwith an additional transform. Fortunately, these goals can beaccomplished with a scheme that is very close to well-known multicarriermodulation techniques, as explained next. We will start with a generalframework for multicarrier modulation and then give examples of OFDM andmulticarrier filter bank implementations.

Let us consider the following components of a time frequency modulation:

-   -   A lattice or grid on the time frequency plane, that is a        sampling of the time axis with sampling period T and the        frequency axis with sampling period Δƒ.        Λ={(nT,mΔƒ),n,mϵ        }  (9)    -   A packet burst with total duration NT secs and total bandwidth        MΔƒ Hz    -   A set of modulation symbols X[n, m], n=0, N−1, m=0, . . . , M−1        we wish to transmit over this burst    -   A transmit pulse g_(tr)(t) with the property of being orthogonal        to translations by T and modulations by Δƒ (generally required        if the receiver uses the same pulse as the transmitter)

$\begin{matrix}{\left\langle {{{\mathcal{g}}_{tr}(t)},{{{\mathcal{g}}_{tr}\left( {t - {nT}} \right)}e^{j\; 2\pi\; m\;\Delta\;{f{({t - {nT}})}}}}} \right\rangle = {{\int{{{\mathcal{g}}_{tr}^{*}(t)}{{\mathcal{g}}_{r}\left( {t - {nT}} \right)}e^{j\; 2\pi\; m\;\Delta\;{f{({t - {nT}})}}}{dt}}} = {{\delta(m)}{\delta(n)}}}} & (10)\end{matrix}$

Given the above components, the time-frequency modulator is a Heisenbergoperator on the lattice Λ, that is, it maps the two dimensional symbolsX[n, m] to a transmitted waveform, via a superposition ofdelay-and-modulate operations on the pulse waveform g_(tr)(t)

$\begin{matrix}{{s(t)} = {\sum\limits_{m = {{- M}\text{/}2}}^{{M\text{/}2} - 1}\;{\sum\limits_{n = 0}^{N - 1}\;{{X\left\lbrack {n,m} \right\rbrack}{{\mathcal{g}}_{tr}\left( {t - {nT}} \right)}e^{j\; 2\pi\; m\;\Delta\;{f{({t - {nT}})}}}}}}} & (11)\end{matrix}$

More formally

where we denote by Π_(X)(⋅) the “discrete” Heisenberg operator,parameterized by discrete values X[n, m].

Notice the similarity of (12) with the channel equation (8). This is notby coincidence, but rather because we apply a modulation effect thatmimics the channel effect, so that the end effect of the cascade ofmodulation and channel is more tractable at the receiver. It is notuncommon practice; for example, linear modulation (aimed at timeinvariant channels) is in its simplest form a convolution of thetransmit pulse g(t) with a delta train of QAM information symbolssampled at the Baud rate T.

$\begin{matrix}{{s(t)} = {\sum\limits_{n = 0}^{N - 1}\;{{X\lbrack n\rbrack}{{\mathcal{g}}\left( {t - {nT}} \right)}}}} & (13)\end{matrix}$

In the present case, aimed at the time varying channel, weconvolve-and-modulate the transmit pulse (c.f. the channel Eq. (2)) witha two dimensional delta train which samples the time frequency domain ata certain Baud rate and subcarrier spacing.

The sampling rate in the time-frequency domain is related to thebandwidth and time duration of the pulse g_(tr)(t); namely, itstime-frequency localization. In order for the orthogonality condition of(10) to hold for a frequency spacing Δƒ, the time spacing must beT≥1/Δƒ. The critical sampling case of T=1/Δƒ is generally not practicaland refers to limiting cases, for example to OFDM systems with cyclicprefix length equal to zero or to filter banks with g_(tr)(t) equal tothe ideal Nyquist pulse.

Some examples illustrate these principles:

Example 1: OFDM Modulation

Let us consider an OFDM system with M subcarriers, symbol lengthT_(OFDM), cyclic prefix length T_(CP) and subcarrier spacing 1/T_(OFDM).If we substitute in Equation (11) symbol duration T=T_(OFDM)+T_(CP),number of symbols N=1, subcarrier spacing Δƒ=1/T_(OFDM) and g_(tr)(t) asquare window that limits the duration of the subcarriers to the symbollength T

$\begin{matrix}{{{\mathcal{g}}_{tr}(t)} = \left\{ \begin{matrix}{{1\text{/}\sqrt{T - T_{CP}}},} & {{- T_{CP}} < t < {T - T_{CP}}} \\{0,} & {else}\end{matrix} \right.} & (14)\end{matrix}$then we obtain the OFDM formula

$\begin{matrix}{{x(t)} = {\sum\limits_{m = {{- M}\text{/}2}}^{{M\text{/}2} - 1}\;{{X\left\lbrack {n,m} \right\rbrack}{{\mathcal{g}}_{tr}(t)}e^{j\; 2\pi\; m\;\Delta\;{ft}}}}} & (15)\end{matrix}$

Technically, the pulse of Eq. (14) is not orthonormal but is orthogonalto the receive filter (where the CP samples are discarded).

Example 2: Single Carrier Modulation

Equation (11) reduces to single carrier modulation if we substitute M=1subcarrier, T equal to the Baud period and g_(tr)(t) equal to a squareroot raised cosine Nyquist pulse.

Example 3: Multicarrier Filter Banks (MCFB)

Equation (11) describes a MCFB if g_(tr)(t) is a square root raisedcosine Nyquist pulse with excess bandwidth α, T is equal to the Baudperiod and Δƒ=(1+α)/T.

Expressing the modulation operation as a Heisenberg transform as in Eq.(12) may be counterintuitive. That is, modulation is usually perceivedas a transformation of the modulation symbols X[m, n] to a transmitwaveform s(t). The Heisenberg transform instead, uses X[m, n] asweights/parameters of an operator that produces s(t) when applied to theprototype transmit filter response g_(tr)(t)—c.f. Eq. (12). Whilecounterintuitive, this formulation is useful in pursuing an abstractionof the modulation-channel-demodulation cascade effects in a twodimensional domain where the channel can be described as time invariant.

Attention is turned next to the processing on the receiver side neededto go back from the waveform domain to the time-frequency domain. Sincethe received signal has undergone the cascade of two Heisenbergtransforms (one by the modulation effect and one by the channel effect),it is natural to inquire what the end-to-end effect of this cascade is.The answer to this question is given by the following result:

Proposition 1: Let two Heisenberg transforms as defined by Eqs. (8), (2)be parametrized by impulse responses h₁(τ, v), h₂(τ, v) and be appliedin cascade to a waveform g(t) ϵ

. ThenΠ_(h) ₂ (Π_(h) ₁ (g(t)))=Π_(h)(g(t))  (16)where h(τ, v)=h₂(τ, v)⊙h₁(τ, v) is the “twisted” convolution of h₁(τ,v), h₂(τ, v) defined by the following convolve-and-modulate operationh(τ,v)=∫∫h ₂(τ′,v′)h ₁(τ−τ′,v−v′)e ^(j2πv′(τ−τ′)t) dτ′dv′  (17)

Applying the above result to the cascade of the modulation and channelHeisenberg transforms of (12) and (8), one can show that the receivedsignal is given by the Heisenberg transformr(t)=Π_(ƒ)(g _(tr)(t))+v(t)=∫∫ƒ(τ,v)e ^(j2πv(t−τ)) g_(tr)(t−τ)dvdτ+v(t)  (18)where v(t) is additive noise and ƒ(τ, v), the impulse response of thecombined transform, is given by the twisted convolution of X[n, m] andh(τ, v)

$\begin{matrix}{{f\left( {\tau,v} \right)} = {{{h\left( {\tau,v} \right)} \odot {X\left\lbrack {n,m} \right\rbrack}} = {\sum\limits_{m = {{- M}\text{/}2}}^{{M\text{/}2} - 1}\;{\sum\limits_{n = 0}^{N - 1}\;{{X\left\lbrack {n,m} \right\rbrack}{h\left( {{\tau - {nT}},{v - {m\;\Delta\; f}}} \right)}e^{j\; 2{\pi{({v - {m\;\Delta\; f}})}}{nT}}}}}}} & (19)\end{matrix}$

This result can be considered an extension of the single carriermodulation case, where the received signal through a time invariantchannel is given by the convolution of the QAM symbols with a compositepulse, that pulse being the convolution of the transmitter pulse and thechannel impulse response.

With this result established we are ready to examine exemplary receiverprocessing steps.

Receiver Processing and the Wigner Transform

Typical communication system design generally requires that the receiverperform a matched filtering operation, taking the inner product of thereceived waveform with the transmitter pulse, appropriately delayed orotherwise distorted by the channel. In the present case, we have used acollection of delayed and modulated transmit pulses, and a matchedfiltering operation is typically performed with respect to each one ofthem.

FIG. 4 provides a conceptual view of this processing. On thetransmitter, we modulate a set of M subcarriers for each symbol wetransmit, while on the receiver we perform matched filtering on each ofthose subcarrier pulses. We define a receiver pulse g_(r)(t) and takethe inner product with a collection of delayed and modulated versions ofit. The receiver pulse g_(r)(t) is in many cases identical to thetransmitter pulse, but we keep the separate notation to cover some caseswhere it is not (most notably in OFDM where the CP samples have to bediscarded).

While this approach will yield the sufficient statistics for datadetection in the case of an ideal channel, a concern can be raised herefor the case of non-ideal channel effects. In this case, the sufficientstatistics for symbol detection are obtained by matched filtering withthe channel-distorted, information-carrying pulses (assuming that theadditive noise is white and Gaussian). In many well designedmulticarrier systems however (e.g., OFDM and MCFB), the channeldistorted version of each subcarrier signal is only a scalar version ofthe transmitted signal, allowing for a matched filter design that isindependent of the channel and uses the original transmitted subcarrierpulse. We will make these statements more precise shortly and examinethe required conditions for this to be true.

In actual embodiments of an OTFS receiver, this matched filtering may beimplemented in the digital domain using an FFT or a polyphase transformfor OFDM and MCFB respectively. However, for purposes of the presentdiscussion, we will consider a generalization of this matched filteringby taking the inner product <g_(r)(t−τ)e^(j2πv(t−τ)), r(t)> of thereceived waveform with the delayed and modulated versions of thereceiver pulse for arbitrary time and frequency offset (τ, v). Whilelikely not necessarily a practical implementation, it allows us to viewthe operations of FIG. 4 as a two dimensional sampling of this moregeneral inner product.

Let us define the inner productA _(g) _(r) _(,r)(τ,v)=<g _(r)(t−τ)e ^(j2πv(t−τ)) ,r(t)>=∫g* _(r)(t−τ)e^(−j2πv(t−τ)) r(t)dt   (20)

The function A_(g) _(r) _(,r)(τ, v) is known as the cross-ambiguityfunction and yields the matched filter output if sampled at τ=nT, v=mΔƒ(on the lattice Λ), i.e.,Y[n,m]=A _(g) _(r) _(,r)(τ,v)|_(τ=nT,v=mΔƒ)  (21)

The ambiguity function is related to the inverse of the Heisenbergtransform, namely the Wigner transform. FIG. 4 provides an intuitivefeel for that, as the receiver appears to invert the operations of thetransmitter. More formally, if we take the cross-ambiguity or thetransmit and receive pulses A_(g) _(r) _(g) _(tr) (τ, v), and use it asthe impulse response of the Heisenberg operator, then we obtain theorthogonal cross-projection operatorΠ_(A) _(gr,gtr) (y(t))=g _(tr)(t)<g _(r)(t),y(t)>

In words, the coefficients that come out of the matched filter, if usedin a Heisenberg representation, will provide the best approximation tothe original y(t) in the sense of minimum square error.

One key question to be addressed is the relationship is between thematched filter output Y[n, m] (or more generally Y(τ, v)) and thetransmitter input X[n,m]. We have already established in (18) that theinput to the matched filter r(t) can be expressed as a Heisenbergrepresentation with impulse response ƒ(τ, v) (plus noise). The output ofthe matched filter then has two contributionsY(τ,v)=A _(g) _(r) _(,r)(τ,v)=A _(g) _(r) _(,Π) _(ƒ) _((g) _(tr)_()+v])(τ,v)=A _(g) _(r) _(,Π) _(ƒ) _((g) _(tr) ₎(τ,v)+A _(g) _(r)_(,v)(τ,v)  (22)

The last term is the contribution of noise, which we will denote V(τ,v)=A_(g) _(r) _(,v)(τ, v). The first term on the right hand side is thematched filter output to the (noiseless) input comprising of asuperposition of delayed and modulated versions of the transmit pulse.We next establish that this term can be expressed as the twistedconvolution of the two dimensional impulse response ƒ(τ, v) with thecross-ambiguity function (or two dimensional cross correlation) of thetransmit and receive pulses.

The following theorem summarizes the key result.

Theorem 1: Fundamental time-frequency domain channel equation). If thereceived signal can be expressed asΠ_(ƒ)(g _(tr)(t))=∫∫ƒ(τ,v)e ^(j2πv(t−τ)) g _(tr)(t−τ)dvdτ  (23)

Then the cross-ambiguity of that signal with the receive pulse g_(tr)(t)can be expressed asA _(g) _(r) _(,Π) _(ƒ) _((g) _(tr) ₎(τ,v)=ƒ(τ,v)⊙A _(g) _(r) _(,g) _(tr)(τ,v)  (24)

Recall from (19) that ƒ(τ, v)=h(τ, v)⊙X[n, m], that is, the compositeimpulse response is itself a twisted convolution of the channel responseand the modulation symbols.

Substituting ƒ(τ, v) from (19) into (22) we obtain the end-to-endchannel description in the time frequency domain

$\begin{matrix}\begin{matrix}{{Y\left( {\tau,v} \right)} = {{A_{{\mathcal{g}}_{r},{\Pi_{r}{({\mathcal{g}}_{tr})}}}\left( {\tau,v} \right)} + {V\left( {\tau,v} \right)}}} \\{= {{{h\left( {\tau,v} \right)} \odot {X\left\lbrack {n,m} \right\rbrack} \odot {A_{{\mathcal{g}}_{r},{\mathcal{g}}_{tr}}\left( {\tau,v} \right)}} + {V\left( {\tau,v} \right)}}}\end{matrix} & (25)\end{matrix}$where V(τ, v) is the additive noise term. Eq. (25) provides anabstraction of the time varying channel on the time-frequency plane. Itstates that the matched filter output at any time and frequency point(τ, v) is given by the delay-Doppler impulse response of the channeltwist-convolved with the impulse response of the modulation operatortwist-convolved with the cross-ambiguity (or two dimensional crosscorrelation) function of the transmit and receive pulses.

Evaluating Eq. (25) on the lattice Λ we obtain the matched filter outputmodulation symbol estimates{circumflex over (X)}[m,n]=Y[n,m]=Y(τ,v)|_(τ=nT,v=mΔƒ)  (26)

In order to get more intuition on Equations (25), (26). let us firstconsider the case of an ideal channel, i.e., h(τ, v)=δ(τ)δ(v). In thiscase by direct substitution we get the convolution relationship

$\begin{matrix}{{Y\left\lbrack {n,m} \right\rbrack} = {{\sum\limits_{m^{\prime} = {{- M}\text{/}2}}^{{M\text{/}2} - 1}\;{\sum\limits_{n^{\prime} = 0}^{N - 1}\;{{X\left\lbrack {n^{\prime},m^{\prime}} \right\rbrack}{A_{{\mathcal{g}}_{r},{\mathcal{g}}_{tr}}\left( {{\left( {n - n^{\prime}} \right)T},{\left( {m - m^{\prime}} \right)\Delta\; f}} \right)}}}} + {V\left\lbrack {m,n} \right\rbrack}}} & (27)\end{matrix}$

In order to simplify Eq. (27) we will use the orthogonality propertiesof the ambiguity function. Since we use a different transmit and receivepulses we will modify the orthogonality condition on the design of thetransmit pulse we stated in (10) to a bi-orthogonality condition

$\begin{matrix}{\left\langle {{{\mathcal{g}}_{tr}(t)},{{{\mathcal{g}}_{r}\left( {t - {nT}} \right)}e^{j\; 2\pi\; m\;\Delta\;{f{({t - {nT}})}}}}} \right\rangle = {{\int{{{\mathcal{g}}_{tr}^{*}(t)}{{\mathcal{g}}_{r}\left( {t - {nT}} \right)}e^{j\; 2\pi\; m\;\Delta\;{f{({t - {nT}})}}}{dt}}} = {{\delta(m)}{\delta(n)}}}} & (28)\end{matrix}$

Under this condition, only one term survives in (27) and we obtainY[n,m]=X[n,m]+V[n,m]  (29)where V[n, m] is the additive white noise. Eq. (29) shows that thematched filter output does recover the transmitted symbols (plus noise)under ideal channel conditions. Of more interest of course is the caseof non-ideal time varying channel effects. We next show that even inthis case, the channel orthogonalization is maintained (no intersymbolor intercarrier interference), while the channel complex gain distortionhas a closed form expression.

The following theorem summarizes the result as a generalization of (29).

Theorem 2: (End-to-end time-frequency domain channel equation):

If h(τ, v) has finite support bounded by (τ_(max), v_(max)) and if A_(g)_(r) _(,g) _(tr) (τ, v)=0 for τϵ(nT−τ_(max), nT+τ_(max)),vϵ(mΔƒ−v_(max), mΔƒ+v_(max)), that is, the ambiguity functionbi-orthogonality property of (28) is true in a neighborhood of each gridpoint (mΔƒ, nT) of the lattice Λ at least as large as the support of thechannel response h(τ, v), then the following equation holdsY[n,m]=H[n,m]X[n,m]H[n,m]=∫∫h(τ,v)e ^(j2πvnT) e ^(−j2π(v+mΔƒ)τ) dvdτ  (30)

If the ambiguity function is only approximately bi-orthogonal in theneighborhood of Λ (by continuity), then (30) is only approximately true.Eq. (30) is a fundamental equation that describes the channel behaviorin the time-frequency domain. It is the basis for understanding thenature of the channel and its variations along the time and frequencydimensions.

Some observations are now in order on Eq. (30). As mentioned before,there is no interference across X[n, m] in either time n or frequency m.

-   -   The end-to-end channel distortion in the modulation domain is a        (complex) scalar that needs to be equalized.    -   If there is no Doppler, i.e. h(τ, v)=h(τ, 0)δ(v), then Eq. (30)        becomes

$\begin{matrix}\begin{matrix}{{Y\left\lbrack {n,m} \right\rbrack} = {{X\left\lbrack {n,m} \right\rbrack}{\int{{h\left( {\tau,0} \right)}e^{{- j}\; 2\pi\; m\;\Delta\; f\;\tau}d\;\tau}}}} \\{= {{X\left\lbrack {n,m} \right\rbrack}{H\left( {0,{m\;\Delta\; f}} \right)}}}\end{matrix} & (31)\end{matrix}$which is the well-known multicarrier result, that each subcarrier symbolis multiplied by the frequency response of the time invariant channelevaluated at the frequency of that subcarrier.

-   -   If there is no multipath, i.e. h(τ, v)=h(0, v)δ(τ), then        Eq. (30) becomes        Y[n,m]=X[n,m]∫h(v,0)e ^(j2πvnT) dτ  (32)

Notice that the fading each subcarrier experiences as a function of timenT has a complicated expression as a weighted superposition ofexponentials. This is a major complication in the design of wirelesssystems with mobility like LTE; it necessitates the transmission ofpilots and the continuous tracking of the channel, which becomes moredifficult the higher the vehicle speed or Doppler bandwidth is.

Some examples of this general framework are provided below.

Example 3: (OFDM Modulation)

In this case the fundamental transmit pulse is given by (14) and thefundamental receive pulse is

$\begin{matrix}{{{\mathcal{g}}_{r}(t)} = \left\{ \begin{matrix}0 & {{- T_{CP}} < t < 0} \\\frac{1}{\sqrt{T - T_{CP}}} & {0 < t < {T - T_{CP}}} \\0 & {else}\end{matrix} \right.} & (33)\end{matrix}$i.e., the receiver zeroes out the CP samples and applies a square windowto the symbols comprising the OFDM symbol. It is worth noting that inthis case, the bi-orthogonality property holds exactly along the timedimension.

Example 4: (MCFB Modulation)

In the case of multicarrier filter banks g_(tr)(t)=g_(r)(t)=g(t). Thereare several designs for the fundamental pulse g(t). A square root raisedcosine pulse provides good localization along the frequency dimension atthe expense of less localization along the time dimension. If T is muchlarger than the support of the channel in the time dimension, then eachsubchannel sees a flat channel and the bi-orthogonality property holdsapproximately.

In summary, one of the two transforms defining OTFS has now beendescribed. Specifically, an explanation has been provided of how thetransmitter and receiver apply appropriate operators on the fundamentaltransmit and receive pulses and orthogonalize the channel according toEq. (30). Examples have also been provided to illustrate how the choiceof the fundamental pulse affects the time and frequency localization ofthe transmitted modulation symbols and the quality of the channelorthogonalization that is achieved. However, Eq. (30) shows that thechannel in this domain, while free of intersymbol interference, suffersfrom fading across both the time and the frequency dimensions via acomplicated superposition of linear phase factors.

In what follows we start from Eq. (30) and describe the second transformthat defines OTFS; we will show how that transform defines aninformation domain where the channel does not fade in either dimension.

The 2D OTFS Transform

Notice that the time-frequency response H[n, m] in (30) is related tothe channel delay-Doppler response h(τ, v) by an expression thatresembles a Fourier transform. However, there are two importantdifferences: (i) the transform is two dimensional (along delay andDoppler) and (ii) the exponentials defining the transforms for the twodimensions have opposing signs. Despite these difficulties, Eq. (30)points in the direction of using complex exponentials as basis functionson which to modulate the information symbols; and only transmit on thetime-frequency domain the superposition of those modulated complexexponential bases. As is discussed below, this approach exploits Fouriertransform properties and effectively translates a multiplicative channelin one Fourier domain to a convolution channel in the other Fourierdomain.

Given the difficulties of Eq. (30) mentioned above, we need to develop asuitable version of Fourier transform and associated sampling theoryresults. Let us start with the following definitions:

Definition 1: Symplectic Discrete Fourier Transform:

Given a square summable two dimensional sequence X[m, n]ϵ

(Λ) we define

$\begin{matrix}\begin{matrix}{{x\left( {\tau,v} \right)} = {\sum\limits_{m,n}{{X\left\lbrack {n,m} \right\rbrack}e^{{- j}\; 2{\pi{({{vnT} - {\tau\; m\;\Delta\; f}})}}}}}} \\{\overset{\Delta}{=}{{SDFT}\left( {X\left\lbrack {n,m} \right\rbrack} \right)}}\end{matrix} & (34)\end{matrix}$

Notice that the above 2D Fourier transform (known as the SymplecticDiscrete Fourier Transform) differs from the more well known CartesianFourier transform in that the exponential functions across each of thetwo dimensions have opposing signs. This is necessary in this case, asit matches the behavior of the channel equation.

Further notice that the resulting x(τ, v) is periodic with periods(1/Δƒ, 1/T). This transform defines a new two dimensional plane, whichwe will call the delay-Doppler plane, and which can represent a maxdelay of 1/Δƒ and a max Doppler of 1/T. A one dimensional periodicfunction is also called a function on a circle, while a 2D periodicfunction is called a function on a torus (or donut). In this case x(τ,v) is defined on a torus Z with circumferences (dimensions) (1/Δƒ, 1/T).

The periodicity of x(τ, v) (or sampling rate of the time-frequencyplane) also defines a lattice on the delay-Doppler plane, which we willcall the reciprocal lattice

$\begin{matrix}{\Lambda^{\bot} = \left\{ {\left( {{m\frac{1}{\Delta\; f}},{n\frac{1}{T}}} \right),n,{m \in {\mathbb{Z}}}} \right\}} & (35)\end{matrix}$

The points on the reciprocal lattice have the property of making theexponent in (34), an integer multiple of 2π.

The inverse transform is given by:

$\begin{matrix}\begin{matrix}{{X\left\lbrack {n,m} \right\rbrack} = {\frac{1}{c}{\int_{0}^{\frac{1}{\Delta\; f}}{\int_{0}^{\frac{1}{T}}{{x\left( {\tau,v} \right)}e^{j\; 2{\pi{({{vnT} - {\tau\; m\;\Delta\; f}})}}}{dvd}\;\tau}}}}} \\{\overset{\Delta}{=}{{SDFT}^{- 1}\left( {x\left( {\tau,v} \right)} \right)}}\end{matrix} & (36)\end{matrix}$

where c=TΔƒ.

We next define a sampled version of x(τ, v). In particular, we wish totake M samples on the delay dimension (spaced at 1/MΔƒ) and N samples onthe Doppler dimension (spaced at 1/NT). More formally, a denser versionof the reciprocal lattice is defined so that Λ^(⊥) ⊂Λ₀ ^(⊥).

$\begin{matrix}{\Lambda_{0}^{\bot} = \left\{ {\left( {{m\frac{1}{M\;\Delta\; f}},{n\frac{1}{NT}}} \right),n,{m \in {\mathbb{Z}}}} \right\}} & (37)\end{matrix}$

-   -   We define discrete periodic functions on this dense lattice with        period (1/Δƒ, 1/T), or equivalently we define functions on a        discrete torus with these dimensions

$\begin{matrix}{Z_{0}^{\bot} = \left\{ {\left( {{m\frac{1}{M\;\Delta\; f}},{n\frac{1}{NT}}} \right),{m = 0},\ldots\mspace{14mu},{M - 1},{n = 0},\ldots\mspace{14mu},{N - 1},} \right\}} & (38)\end{matrix}$

These functions are related via Fourier transform relationships todiscrete periodic functions on the lattice Λ, or equivalently, functionson the discrete torusZ ₀={(nT,mΔƒ), m=0, . . . ,M−1, n=0, . . . N−1,}  (39)

We wish to develop an expression for sampling Eq. (34) on the lattice of(38). First, we start with the following definition.

Definition 2: Symplectic Finite Fourier Transform:

If X_(p) [k, l] is periodic with period (N, M), then we define

$\begin{matrix}\begin{matrix}{{x_{p}\left\lbrack {m,n} \right\rbrack} = {\sum\limits_{k = 0}^{N - 1}\;{\sum\limits_{l = {- \frac{M}{2}}}^{\frac{M}{2} - 1}\;{{X_{p}\left\lbrack {k,l} \right\rbrack}e^{{- j}\; 2{\pi{({\frac{nk}{N} - \frac{ml}{M}})}}}}}}} \\{\overset{\Delta}{=}{{SFFT}\left( {X\left\lbrack {k,l} \right\rbrack} \right)}}\end{matrix} & (40)\end{matrix}$

Notice that x_(p)[m, n] is also periodic with period [M, N] or,equivalently, it is defined on the discrete torus Z₀ ^(⊥). Formally, theSFFT(X[n, m]) is a linear transformation from

(Z₀)→

(Z₀ ^(⊥)).

Let us now consider generating x_(p)[m, n] as a sampled version of (34),i.e.,

${x_{p}\left\lbrack {m,n} \right\rbrack} = {{x\left\lbrack {m,n} \right\rbrack} = \left. {x\left( {\tau,v} \right)} \middle| {}_{{\tau = \frac{m}{M\;\Delta\; f}},{v = \frac{n}{NT}}}. \right.}$Then we can show that (40) still holds where X_(p)[m, n] is aperiodization of X[n, m] with period (N, M)

$\begin{matrix}{{X_{p}\left\lbrack {n,m} \right\rbrack} = {\sum\limits_{l,{k = {- \infty}}}^{\infty}\;{X\left\lbrack {{n - {kN}},{m - {lM}}} \right\rbrack}}} & (41)\end{matrix}$

This is similar to the result that sampling in one Fourier domaincreates aliasing in the other domain.

The inverse discrete (symplectic) Fourier transform is given by

$\begin{matrix}\begin{matrix}{{X_{p}\left\lbrack {n,m} \right\rbrack} = {\frac{1}{MN}{\sum\limits_{l,k}{{x\left\lbrack {l,k} \right\rbrack}e^{j\; 2{\pi{({\frac{nk}{N} - \frac{ml}{M}})}}}}}}} \\{\overset{\Delta}{=}{{SFFT}^{- 1}\left( {x\left\lbrack {l,k} \right\rbrack} \right)}}\end{matrix} & (42)\end{matrix}$where l=0, . . . , M−1, k=0, . . . , N−1. If the support of X[n, m] istime-frequency limited to Z₀ (no aliasing in (41)), then X_(p)[n,m]=X[n, m] for n, m ϵ Z₀, and the inverse transform (42) recovers theoriginal signal.

The SDFT is termed “discrete” because it represents a signal using adiscrete set of exponentials, while the SFFT is called “finite” becauseit represents a signal using a finite set of exponentials.

In the present context, an important property of the symplectic Fouriertransform is that it transforms a multiplicative channel effect in onedomain to a circular convolution effect in the transformed domain. Thisis summarized in the following proposition:

Proposition 2: Let X₁[n, m]ϵ

(Z₀), X₂[n, m]ϵ

(Z₀) be periodic 2D sequences. ThenSFFT(X ₁ [n,m]*X ₂ [n,m])=SFFT(X ₁ [n,m])·SFFT(X ₂ [n,m])  (43)where * denotes two dimensional circular convolution. With thisframework established we are ready to define the OTFS modulation.

Discrete OTFS Modulation:

Consider a set of NM QAM information symbols arranged on a 2D grid x[l,k], k=0, . . . , N−1, l=0, . . . , M−1 we wish to transmit. We willconsider x[l, k] to be two dimensional periodic with period [N, M].Further, assume a multicarrier modulation system defined by

-   -   A lattice on the time frequency plane, that is a sampling of the        time axis with sampling period T and the frequency axis with        sampling period Δƒ (c.f. Eq. (9)).    -   A packet burst with total duration NT secs and total bandwidth        MΔƒ Hz.    -   Transmit and receive pulses g_(tr)(t), g_(tr)(t)ϵ L₂(        ) satisfying the bi-orthogonality property of (28)    -   A transmit windowing square summable function W_(tr)[n, m]ϵ        (Λ) multiplying the modulation symbols in the time-frequency        domain    -   A set of modulation symbols X[n, m], n=0, . . . , N−1, m=0, . .        . , M−1 related to the information symbols x[k, l] by a set of        basis functions b_(k,l)[n, m]

$\begin{matrix}{{{X\left\lbrack {n,m} \right\rbrack} = {\frac{1}{MN}{W_{tr}\left\lbrack {n,m} \right\rbrack}{\sum\limits_{k = 0}^{N - 1}\;{\sum\limits_{l = 0}^{M - 1}\;{{x\left\lbrack {l,k} \right\rbrack}{b_{k,l}\left\lbrack {n,m} \right\rbrack}}}}}}{{b_{k,l}\left\lbrack {n,m} \right\rbrack} = e^{j\; 2{\pi{({\frac{ml}{M} - \frac{nk}{N}})}}}}} & (44)\end{matrix}$

-   -   -   where the basis functions b_(k,l)[n, m] are related to the            inverse symplectic Fourier transform (c.f., Eq. (42))

Given the above components, we define the discrete OTFS modulation viathe following two stepsX[n,m]=W _(tr) [n,m]SFFT ⁻¹(x[k,l])s(t)=Π_(X)(g _(tr)(t))  (45)

The first equation in (45) describes the OTFS transform, which combinesan inverse symplectic transform with a widowing operation. The secondequation describes the transmission of the modulation symbols X[n, m]via a Heisenberg transform of g_(tr)(t) parameterized by X[n, m]. Moreexplicit formulas for the modulation steps are given by Equations (42)and (11).

While the expression of the OTFS modulation via the symplectic Fouriertransform reveals important properties, it is easier to understand themodulation via Eq. (44), that is, transmitting each information symbolx[k, l] by modulating a 2D basis function b_(k,l)[n, m] on thetime-frequency plane.

Discrete OTFS Demodulation:

Let us assume that the transmitted signal s(t) undergoes channeldistortion according to (8), (2) yielding r(t) at the receiver. Further,let the receiver employ a receive windowing square summable functionW_(r)[n, m]. Then, the demodulation operation consists of the followingsteps:

-   -   (i) Matched filtering with the receive pulse, or more formally,        evaluating the ambiguity function on Λ (Wigner transform) to        obtain estimates of the time-frequency modulation symbols        Y[n,m]=A _(g) _(r) _(,y)(τ,v)|_(τ=nT,v=mΔƒ)  (46)    -   (ii) windowing and periodization of Y[n, m]

$\begin{matrix}{{{Y_{w}\left\lbrack {n,m} \right\rbrack} = {{W_{r}\left\lbrack {n,m} \right\rbrack}{Y\left\lbrack {n,m} \right\rbrack}}}{{Y_{p}\left\lbrack {n,m} \right\rbrack} = {\sum\limits_{k,{l = {- \infty}}}^{\infty}\;{Y_{w}\left\lbrack {{n - {kN}},{m - {lM}}} \right\rbrack}}}} & (47)\end{matrix}$

-   -   (iii) and applying the symplectic Fourier transform on the        periodic sequence Y_(p)[n, m]        {circumflex over (x)}[l,k]=y[l,k]=SFFT(Y _(p) [n,m])  (48)

The first step of the demodulation operation can be interpreted as amatched filtering operation on the time-frequency domain as we discussedearlier. The second step is there to ensure that the input to the SFFTis a periodic sequence. If the trivial window is used, this step can beskipped. The third step can also be interpreted as a projection of thetime-frequency modulation symbols on the orthogonal basis functions

$\begin{matrix}{{{\hat{x}\left\lbrack {l,k} \right\rbrack} = {\sum\limits_{m = 0}^{M - 1}\;{\sum\limits_{n = 0}^{N - 1}\;{{\hat{X}\left( {n,m} \right)}{b_{k,l}^{*}\left( {n,m} \right)}}}}}{{b_{k,l}^{*}\left( {n,m} \right)} = e^{{- j}\; 2{\pi{({\frac{lm}{L} - \frac{kn}{K}})}}}}} & (49)\end{matrix}$

The discrete OTFS modulation defined above points to efficientimplementation via discrete-and-periodic FFT type processing. However,it potentially does not provide insight into the time and bandwidthresolution of these operations in the context of two dimensional Fouriersampling theory. We next introduce continuous OTFS modulation and relatethe more practical discrete OTFS as a sampled version of the continuousmodulation.

Continuous OTFS Modulation:

Consider a two dimensional periodic function x(τ, v) with period [1/Δƒ,1/T] that we wish to transmit. The choice of the period may seemarbitrary at this point, but the rationale for its choice will becomeapparent after the discussion below. Further, assume a multicarriermodulation system defined by

-   -   A lattice on the time frequency plane, that is a sampling of the        time axis with sampling period T and the frequency axis with        sampling period Δƒ (c.f. Eq. (9)).    -   Transmit and receive pulses g_(tr)(t), g_(tr)(t)ϵL₂(        ) satisfying the bi-orthogonality property of (28)    -   A transmit windowing function W_(tr)[n, m]ϵ        (Λ) multiplying the modulation symbols in the time-frequency        domain

Given the above components, we define the continuous OTFS modulation viathe following two stepsX[n,m]=W _(tr) [n,m]SDFT ⁻¹(x(τ,v))s(t)=Π_(X)(g _(tr)(t))  (50)

The first equation describes the inverse discrete time-frequencysymplectic Fourier transform [c.f. Eq. (36)] and the windowing function,while the second equation describes the transmission of the modulationsymbols via a Heisenberg transform [c.f. Eq. (11)].

Continuous OTFS Demodulation:

Let us assume that the transmitted signal s(t) undergoes channeldistortion according to (8), (2) yielding r(t) at the receiver. Further,let the receiver employ a receive windowing function W_(r)[n, m]ϵ

(Λ). Then, the demodulation operation consists of two steps:

-   -   (i) Evaluating the ambiguity function on Λ (Wigner transform) to        obtain estimates of the time-frequency modulation symbols        Y[n,m]=A _(g) _(r) _(,y)(τ,v)|_(τ=nT,v=mΔƒ)  (51)    -   (ii) Windowing and applying the symplectic Fourier transform on        the modulation symbols        {circumflex over (x)}(τ,v)=SDFT(W _(r) [n,m]Y[n,m])  (52)

Notice that in (51), (52) there is no periodization of Y[n, m], sincethe SDFT is defined on aperiodic square summable sequences. Theperiodization step needed in discrete OTFS can be understood as follows.Suppose we wish to recover the transmitted information symbols byperforming a continuous OTFS demodulation and then sampling on thedelay-Doppler grid

${\hat{x}\left( {l,k} \right)} = \left. {\hat{x}\left( {\tau,v} \right)} \right|_{{\tau = \frac{m}{M\;\Delta\; f}},{v = \frac{n}{NT}}}$

Since performing a continuous symplectic Fourier transform is generallynot practical we consider whether the same result can be obtained usingSFFT. The answer is that SFFT processing will produce exactly thesamples we are looking for if the input sequence is first periodized(aliased). See also equation (40) and (41).

We have now described each of the steps of an exemplary form of OTFSmodulation. We have also discussed how the Wigner transform at thereceiver inverts the Heisenberg transform at the transmitter [c.f. Eqs.(27), (29)], and similarly for the forward and inverse symplecticFourier transforms.

FIG. 5 illustratively represents an exemplary embodiment of OTFSmodulation, including the transformation of the time-frequency plane tothe Doppler-delay plane. In addition, FIG. 5 indicates relationshipsbetween sampling rate, delay resolution and time resolution. Referringto FIG. 5, in a first operation a Heisenberg transform translates atime-varying convolution channel in the waveform domain to an orthogonalbut still time varying channel in the time frequency domain. For a totalbandwidth BW and M subcarriers the frequency resolution is Δƒ=BW/M. Fora total frame duration T_(ƒ) and N symbols the time resolution isT=T_(ƒ)/N.

In a second operation a SFFT transform translates the time-varyingchannel in the time-frequency domain to a time invariant one in thedelay-Doppler domain. The Doppler resolution is 1/T_(ƒ) and the delayresolution is 1/BW. The choice of window can provide a tradeoff betweenmain lobe width (resolution) and side lobe suppression, as in classicalspectral analysis.

Channel Equation in the OTFS Domain

A mathematical characterization of the end-to-end signal relationship inan OTFS system when a non-ideal channel is between the transmitter andreceiver will now be provided. Specifically, this section demonstrateshow the time varying channel in (2), (8), is transformed to a timeinvariant convolution channel in the delay Doppler domain.

Proposition 3: Consider a set of NM QAM information symbols arranged ina 2D periodic sequence x[l, k] with period [M, N]. The sequence x[k, l]undergoes the following transformations:

-   -   It is modulated using the discrete OTFS modulation of Eq. (45).    -   It is distorted by the delay-Doppler channel of Eqs. (2), (8).    -   It is demodulated by the discrete OTFS demodulation of Eqs.        (46), (48).

The estimated sequence {circumflex over (x)}[l, k] obtained afterdemodulation is given by the two dimensional periodic convolution

$\begin{matrix}{{\hat{x}\left\lbrack {l,k} \right\rbrack} \simeq {\frac{1}{MN}{\sum\limits_{m = 0}^{M - 1}\;{\sum\limits_{n = 0}^{N - 1}\;{{x\left\lbrack {m,n} \right\rbrack}{h_{w}\left( {\frac{l - m}{M\;\Delta\; f},\frac{k - n}{NT}} \right)}}}}}} & (53)\end{matrix}$

-   -   of the input QAM sequence x[m, n] and a sampled version of the        windowed impulse response h_(w)(⋅),

$\begin{matrix}{{h_{w}\left( {\frac{l - m}{M\;\Delta\; f},\frac{k - n}{NT}} \right)} = \left. {h_{w}\left( {\tau^{\prime},v^{\prime}} \right)} \right|_{{\tau^{\prime} = \frac{l - m}{M\;\Delta\; f}},{v^{\prime} = \frac{k - n}{NT}}}} & (54)\end{matrix}$

-   -   where h_(w)(τ′, v′) denotes the circular convolution of the        channel response with a windowing function        h _(w)(τ′,v′)=∫∫e ^(−j2πvτ) h(τ,v)w(τ′−τ,v′−v)dτdv  (55)

To be precise, the window w(τ, v) is circularly convolved with aslightly modified version of the channel impulse response e^(−j2πvτ)h(τ,v) (by a complex exponential) as can be seen in the equation. Thewindowing function w(τ, v) is the symplectic Fourier transform of thetime-frequency window W[n, m]

$\begin{matrix}{{w\left( {\tau,v} \right)} = {\sum\limits_{m = 0}^{M - 1}\;{\sum\limits_{n = 0}^{N - 1}\;{{W\left\lbrack {n,m} \right\rbrack}e^{{- j}\; 2{\pi{({{vnT} - {\tau\; m\;\Delta\; f}})}}}}}}} & (56)\end{matrix}$

-   -   and where W[n, m] is the product of the transmit and receive        window.        W[n,m]=W _(tr) [n,m]W _(r) [n,m]  (57)

In many cases, the windows in the transmitter and receiver are matched,i.e., W_(tr)[n, m]=W₀[n, m] and W_(r)[n, m]=W*₀[n, m], hence W[n,m]=|W₀[n, m]|².

The window effect is to produce a blurred version of the originalchannel with a resolution that depends on the span of the frequency andtime samples available. If one considers the rectangular (or trivial)window, i.e., W[n, m]=1, n=0, . . . , N−1, m=−M/2, . . . , M/2−1 andzero else, then its SDFT w(τ, v) in (56) is the two dimensionalDirichlet kernel with bandwidth inversely proportional to N and M.

There are several other uses of the window function. The system can bedesigned with a window function aimed at randomizing the phases of thetransmitted symbols. This randomization may be more important for pilotsymbols than data carrying symbols. For example, if neighboring cellsuse different window functions, the problem of pilot contamination isavoided.

Channel Estimation in the OTFS Domain

There is a variety of different ways a channel estimation scheme couldbe designed for an OTFS system, and a variety of differentimplementation options and details

A straightforward way to perform channel estimation entails transmittinga sounding OTFS frame containing a discrete delta function in the OTFSdomain or, equivalently, a set of unmodulated carriers in the timefrequency domain. From a practical standpoint, the carriers may bemodulated with known, say BPSK, symbols which are removed at thereceiver, as is common in many OFDM systems. FIG. 6 shows a discreteimpulse in the OTFS domain which may be used for purposes of channelestimation.

However, this approach may be wasteful as the extent of the channelresponse is only a fraction of the full extent of the OTFS frame (1/T,1/Δƒ). For example, in LTE systems 1/T≈15 KHz while the maximum Dopplershift ƒ_(d,max) is typically one to two orders of magnitude smaller.Similarly 1/Δƒ≈67 usec, while maximum delay spread τ_(max) is again oneto two orders of magnitude less. We therefore can have a much smallerregion of the OTFS frame devoted to channel estimation while the rest ofthe frame carries useful data. More specifically, for a channel withsupport (±ƒ_(d,max), ±τ_(max)) we need an OTFS subframe of length(2ƒ_(d,max)/T, 2τ_(max)/Δƒ).

In the case of multiuser transmission, each UE can have its own channelestimation subframe positioned in different parts of the OTFS frame.However, this channel estimation subframe may be relatively limited insize. For example, if τ_(max) is 5% of the extent of the delay dimensionand ƒ_(d,max) is 5% of the Doppler dimension, the channel estimationsubframe need only be 5%×5%=0.25% of the OTFS frame.

Importantly, although the channel estimation symbols are limited to asmall part of the OTFS frame, they actually sound the wholetime-frequency domain via the corresponding two-dimensionaltime-frequency basis functions associated with these symbols.

A different approach to channel estimation is to devote pilot symbols ona subgrid in the time-frequency domain. The key question in thisapproach is the determination of the density of pilots that issufficient for channel estimation without introducing aliasing. Assumethat the pilots occupy the subgrid (n₀T, m₀Δƒ) for some integers n₀, m₀.Recall that for this grid the SDFT will be periodic with period (1/n₀T,1/m₀Δƒ). Then, applying the aliasing results discussed earlier to thisgrid, we obtain an alias-free Nyquist channel support region of(±ƒ_(d,max), ±τ_(max))=(±1/2n₀T, ±1/2m₀Δƒ). The density of the pilotscan then be determined from this relation given the maximum support ofthe channel. The pilot subgrid should extend to the whole time-frequencyframe, so that the resolution of the channel is not compromised.

OTFS Access: Multiplexing More than One User

There are a variety of ways to multiplex several uplink or downlinktransmissions in one OTFS frame. Here we will briefly review thefollowing multiplexing methods:

-   -   Multiplexing in the OTFS delay-Doppler domain    -   Multiplexing in the time-frequency domain    -   Multiplexing in the code spreading domain    -   Multiplexing in the spatial domain

1. Multiplexing in the Delay-Doppler Domain:

This is potentially the most natural multiplexing scheme for downlinktransmissions. Different sets of OTFS basis functions, or sets ofinformation symbols or resource blocks, are given to different users.Given the orthogonality of the basis functions, the users can beseparated at the UE receiver. The UE need only demodulate the portion ofthe OTFS frame that is assigned to it.

In contrast to conventional communication systems, in an OTFS systemeven a small subframe or resource block in the OTFS domain will betransmitted over the whole time-frequency frame via two-dimensionalbasis functions and will experience the average channel response. FIG. 7illustrates this point by showing two different basis functionsbelonging to different users. Because of this, there is no compromise onchannel resolution for each user, regardless of the resource block orsubframe size.

In the uplink direction, transmissions from different users experiencedifferent channel responses. Hence, the different subframes in the OTFSdomain will experience a different convolution channel. This canpotentially introduce inter-user interference at the edges where twouser subframes are adjacent, and would require guard gaps to eliminateit. In order to avoid this overhead, a different multiplexing scheme canbe used in the uplink as explained next.

2. Multiplexing in the Time-Frequency Domain:

In this approach, resource blocks or subframes are allocated todifferent users in the time-frequency domain. FIG. 8 illustrates thisfor a three user case. As shown in FIG. 8, a first user (U1) occupiesthe whole frame length but only half the available subcarriers. A seconduser (U2) and a third user (U3) occupy the other half subcarriers, anddivide the total length of the frame between them.

Notice that in this case, each user employs a slightly different versionof the OTFS modulation described. One difference is that each user iperforms an SFFT on a subframe (N_(i), M_(i)), N_(i)≤N, M_(i)≤M. Thisreduces the resolution of the channel, or in other words reduces theextent of the time-frequency plane in which each user will experienceits channel variation. On the other side, this also gives the schedulerthe opportunity to schedule users in parts of the time-frequency planewhere their channel is best.

If it is desired to extract the maximum diversity of the channel andallocate users across the whole time-frequency frame, users can bemultiplexed via interleaving. In this case, one user occupies asubsampled grid of the time-frequency frame, while another user occupiesanother subsampled grid adjacent to it. FIG. 9 shows the same threeusers as in FIG. 8, but interleaved on the subcarrier dimension. Ofcourse, interleaving is possible in the time dimension as well, and/orin both dimensions. The degree of interleaving, or subsampling the gridper user is only limited by the spread of the channel that must beaccommodated.

3. Multiplexing in the Time Frequency Spreading Code Domain:

Assume that it is desired to design a random access PHY and MAC layerwhere users can access the network without having to undergo elaborateRACH and other synchronization procedures. There is a perceived need forsuch a system to support Internet of Things (IoT) deployments. OTFS cansupport such a system by assigning each user a different two-dimensionalwindow function that is designed as a randomizer. In this embodiment thewindows of different users are designed to be nearly orthogonal to eachother and nearly orthogonal to time and frequency shifts. Each user thenonly transmits on one or a few basis functions and uses the window as ameans to randomize interference and provide processing gain. This canresult in a much simplified system that may be attractive for low cost,short burst type of IoT applications.

4. Multiplexing in the Spatial Domain:

Finally, like other OFDM multicarrier systems, a multi-antenna OTFSsystem can support multiple users transmitting on the same basisfunctions across the whole time-frequency frame. The users are separatedby appropriate transmitter and receiver beamforming operations.

Exemplary Implementations of OTFS Communication Systems

As discussed above, embodiments of Orthogonal Time Frequency Space(OTFS) modulation are comprised of a cascade of two transformations. Thefirst transformation maps the two dimensional plane where theinformation symbols reside (and which may be termed the delay-Dopplerplane) to the time frequency plane. The second transformation transformsthe time frequency domain to the waveform time domain where thetransmitted signal is actually constructed. This transform can bethought of as a generalization of multicarrier modulation schemes.

FIG. 10 illustrates components of an exemplary OTFS transceiver 1000.The OTFS transceiver 1000 can be used as one or both of the exemplaryOTFS transceivers 315 illustrated in the communication system 300 ofFIG. 3A. The OTFS transceiver 1000 includes a transmitter module 1005that includes a pre-equalizer 1010, an OTFS encoder 1020 and an OTFSmodulator 1030. The OTFS transceiver 1000 also includes a receivermodule 1055 that includes a post-equalizer 1080, an OTFS decoder 1070and an OTFS demodulator 1060. The components of the OTFS transceiver maybe implemented in hardware, software, or a combination thereof. For ahardware implementation, the processing units may be implemented withinone or more application specific integrated circuits (ASICs), digitalsignal processors (DSPs), digital signal processing devices (DSPDs),programmable logic devices (PLDs), field programmable gate arrays(FPGAs), processors, controllers, micro-controllers, microprocessors,other electronic units designed to perform the functions describedabove, and/or a combination thereof. The disclosed OTFS methods will bedescribed in view of the various components of the transceiver 1000.

Referring again to FIG. 3A, in one aspect a method of OTFS communicationinvolves transmitting at least one frame of data ([D]) from thetransmitting device 310 to the receiving device 330 through thecommunication channel 320, such frame of data comprising a matrix of upto N² data elements, N being greater than 1. The method comprisesconvolving, within the OTFS transceiver 315-1, the data elements of thedata frame so that the value of each data element, when transmitted, isspread over a plurality of wireless waveforms, each waveform having acharacteristic frequency, and each waveform carrying the convolvedresults from a plurality of said data elements from the data frame [D].Further, during the transmission process, cyclically shifting thefrequency of this plurality of wireless waveforms over a plurality oftimes so that the value of each data element is transmitted as aplurality of cyclically frequency shifted waveforms sent over aplurality of times. At the receiving device 330, the OTFS transceiver315-2 receives and deconvolves these wireless waveforms therebyreconstructing a replica of said at least one frame of data [D]. In theexemplary embodiment the convolution process is such that an arbitrarydata element of an arbitrary frame of data ([D]) cannot be guaranteed tobe reconstructed with full accuracy until substantially all of thesewireless waveforms have been transmitted and received.

FIG. 11 illustrates a comparison of bit error rates (BER) predicted by asimulation of a TDMA system and an OTFS system. Both systems utilize a16 QAM constellation. The simulation modeled a Doppler spread of 100 Hzand a delay spread of 3 microsec. As can be seen from the graphs, theOTFS system offers much lower BER than the TDMA system for the samesignal-to-noise ratio (SNR).

Attention is now directed to FIG. 12, which is a flowchartrepresentative of the operations performed by an OTFS transceiver 1200which may be implemented as, for example, the OTFS transceiver 1000(FIG. 10). The OTFS transceiver 1200 includes a transmitter including amodulator 1210 and a receiver including a demodulator 1220 andtwo-dimensional equalizer 1230. In operation, a transmitter of the OTFStransceiver 1200 receives a two-dimensional symbol stream in the form ofan N×N matrix of symbols, which may hereinafter be referred to as atime-frequency (TF) matrix:xϵC ^(N×N)

As is illustrated in FIG. 13, in one embodiment the modulator 1210functions as an orthogonal map disposed to transform the two-dimensionalTF matrix to the following transmitted waveform:ϕ_(i) M(x)=Σx(i,j)ϕ_(i,j)ϕ_(i,j)⊥ϕ_(k,j)

Referring to FIG. 14, the demodulator 1220 transforms the receivedwaveform to a two-dimensional TF matrix in accordance with an orthogonalmap in order to generate an output stream:ϕ_(r)

y=D(ϕ_(r))

In one embodiment the OTFS transceiver 1200 may be characterized by anumber of variable parameters including, for example, delay resolution(i.e., digital time “tick” or clock increment), Doppler resolution,processing gain factor (block size) and orthonormal basis function. Eachof these variable parameters may be represented as follows.

Delay resolution (digital time tick):

${\Delta\; T} \in {R^{> 0}\mspace{14mu}\left( {{\Delta\; T} = \frac{1}{Bw}} \right)}$

Doppler resolution:

${\Delta\; F} \in {R^{> 0}\mspace{14mu}\left( {{\Delta\; F} = \frac{1}{Trans}} \right)}$

Processing gain factor (block size):N>0

Orthonormal basis of C^(N×1)(spectral shapes):U={u ₁ ,u ₂ , . . . ,u _(N)}

As is illustrated by FIG. 12, during operation the modulator 1210 takesa TF matrix xϵC^(N) ^(×) ^(N) and transforms it into a pulse waveform.In one embodiment the pulse waveform comprises a pulse train defined interms of the Heisenberg representation and the spectral shapes:

$\phi_{t} = {{M(x)} = \left( {\underset{\underset{b_{1}}{︸}}{{\Pi(x)}u_{1}},\underset{\underset{b_{2}}{︸}}{{\Pi(x)}u_{2}},\ldots\mspace{14mu},\underset{\underset{b_{N}}{︸}}{{\Pi(x)}u_{N}}} \right)}$

-   -   where b₁, b₂ . . . b_(N) are illustrated in FIG. 15 and where,        in accordance with the Heisenberg relation:        Π(h*x)=Π(h)·Π(x) in particular:        Π(δ_((x,n)) *x)=L _(t)·Π(x)        Π(δ_((0,w)) *x)=M _(w)·Π(x)

The Heisenberg representation provides that:

${\Pi\text{:}C^{N \times N}}\overset{N}{\rightarrow}{C^{N \times N}\mspace{14mu}{given}\mspace{14mu}{by}\text{:}}$${{\Pi(x)} = {\sum\limits_{{\tau\text{:}w} = 0}^{N - 1}\;{{x\left( {\tau,w} \right)}M_{w}L_{\tau}}}},{x \in C^{N \times N}}$

-   -   where L_(t) and M_(w) are respectively representative of cyclic        time and frequency shifts and may be represented as:

L_(τ) ∈ C^(N × N):  L_(τ)(φ)(t) = φ(t + τ), τ = 0, …  , N − 1${{M_{w} \in {{C^{N \times N} \cdot {M_{w}(\varphi)}}(t)}} = {e^{\frac{j\; 2\pi}{N}{wt}}{\varphi(t)}}},{w = 0},\ldots\mspace{14mu},{N - 1}$

The demodulator 1220 takes a received waveform and transforms it into aTF matrix yϵC^(N) ^(×) ^(N) defined in terms of the Wigner transform andthe spectral shapes:

ϕ_(r) = (b₁, b₂, …  , b_(N))${y\left( {\tau,w} \right)} = {{{D\left( \phi_{r} \right)}\left( {\tau,w} \right)} = \overset{\underset{︷}{{Wigner}\mspace{14mu}{transform}}}{\frac{1}{N}{\sum\limits_{n = 1}^{N}\;\left\langle {{M_{w}L_{\tau}u_{n}},b_{n}} \right\rangle}}}$

Main property of M and D (Stone von Neumann theorem):D(h ^(o) M(x))=h*x where:h(τ,w)≈a(τΔT,wΔF)

As illustrated in FIG. 16, the equalizer 1230 may be implemented as atwo-dimensional decision feedback equalizer configured to perform aleast means square (LMS) equalization procedure such that:y

{circumflex over (x)}Transmitter Grid and Receiver Bin Structure

Attention is now directed to FIGS. 17A-17D, which depict an OTFStransmitter 102 and receiver 104 to which reference will be made indescribing the transmission and reception of OTFS waveforms. Morespecifically, FIGS. 17B-17D illustrate the transmission and reception ofOTFS waveforms relative to a time-frequency transmitter grid or seriesof bins and a corresponding time-frequency receiver grid or series ofbins. As will be discussed below, the receiver 104 will generallyoperate with respect to a time-frequency receive grid of a finer meshthan that of the time-frequency transmit grid associated with thetransmitter 102.

Turning now to FIG. 17A, the transmitter 102 and receiver 104 areseparated by an impaired wireless data channel 100 including one or morereflectors 106. As shown, the reflectors 106 may reflect or otherwiseimpair waveforms (112, 114 a, 114 b) as they travel through the datachannel 100. These reflectors may be inherently represented by thetwo-dimensional (2D) channel state of the channel 100 (see, e.g., thefinite channel h_(eqv,f) of FIG. 18).

In one embodiment the transmitter 102 includes a transmitter processor102 p to package input data into at least one N×M array of data symbols.An encoding process is then used to transmit this array of data symbolsin accordance with the OTFS modulation techniques described herein. Thetransmitted OTFS waveforms are received by a receiver 104, whichincludes a receiver processor 104 p. In one embodiment the receiverprocessor 104 p utilizes information pertaining to the 2D state of thechannel 100 to enable these OTFS waveforms to be decoded and recover thetransmitted data symbols. Specifically, the receiver processor 104 p mayuse an inverse of the OTFS encoding process to decode and extract thisplurality of data symbols. Alternatively the correction of signals fordata channel impairments can be done after the receiver has decoded andextracted the plurality of data symbols.

In some embodiments OTFS data transmission may be implemented bytransforming the input N×M array of data symbols into at least one blockor array of filtered OFDM symbols. This can be done, for example, usingone dimensional Fourier transforms and a filtering process or algorithm.This block or array of filtered OFDM symbols may then be transformedinto at least one block or array of OFTS symbols using various types oftwo dimensional Fourier transforms. These results will typically bestored in transmitter memory 102 m. The stored results can then becommunicated over wireless frequency sub-bands by various methods. Forexample, in one embodiment a transmitter 102 c that employs a series ofM narrow-band filter banks may be utilized. In this implementation thetransmitter 102 c produces a series of M mutually orthogonal waveformstransmitted over at least N time intervals.

In one embodiment gaps or “guard bands” in both time and frequency maybe imposed to minimize the possibility of inadvertent cross talk betweenthe various narrow-band filters and time intervals prior totransmission. Depending on the characteristics of the data channel, anysuch gaps or guard bands can be increased or decreased or set to zero assituations warrant.

Alternatively, the OTFS encoding process may encode the N×M array ofdata symbols onto a manifold compatible with symplectic analysis. Thesymbols may be distributed over a column time axis of length T and rowfrequency axis of length F, thereby producing at least one informationmanifold for storage in transmitter memory 102 m.

The information manifold effectively holds information corresponding tothe input data symbols in a form enabling them to be subsequentlytransformed in accordance with the desired OTFS transformation operationsuch as, for example, a symplectic 2D Fourier transform, a discretesymplectic 2D Fourier transform, a finite symplectic Fourier transform,and the like. In certain embodiments the data symbols may also be spreadprior to being held within an information manifold.

The OTFS processor 102 p may then transform the information manifoldaccording to a 2D symplectic Fourier transform. This transformation maybe effected using any of the previously discussed symplectic 2D Fouriertransforms, discrete symplectic 2D Fourier transforms, and finitesymplectic Fourier transforms. This operation produces at least one 2DFourier transformed information manifold, which may be stored intransmitter memory 102 m.

The OTFS transmitter 102 c will typically transmit this at least one 2DFourier transformed information manifold as a series of “M” simultaneousnarrow band waveforms, each series over consecutive time intervals,until the entire 2D Fourier transformed information manifold has beentransmitted. For example, the transmitter processor 102 p can operate,often on a one column at a time basis, over all frequencies and times ofthis 2D Fourier transformed information manifold. The transmitterprocessor 102 p can select a given column at location n (where n canvary from 1 to N) and transmit a column with a width according to a timeslice of duration proportional to Tμ, where μ=1/N. Those frequencies inthe column slice of this 2D Fourier transformed information manifold(e.g. frequencies corresponding to this transmitting time slice) maythen be passed through a bank of at least M different, nonoverlapping,narrow-band frequency filters. This produces M mutually orthogonalwaveforms. The processor 102 p can then cause these resulting filteredwaveforms to be transmitted, over different transmitted time intervals(e.g. one column at a time), as a plurality of at least M mutuallyorthogonal waveforms until an entire 2D Fourier transformed informationmanifold has been transmitted.

In one embodiment gaps or “guard bands” in both time and frequency maybe imposed to minimize the possibility of inadvertent cross talk betweenthe various narrow-band filters and time intervals prior totransmission. Depending on the characteristics of the data channel, anysuch gaps or guard bands can be increased or decreased or set to zero assituations warrant.

Each OTFS receiver 104 may then receive a channel-convoluted version ofthe 2D Fourier transformed information manifold transmitted by thetransmitter 102. Due to distortions introduced by the channel 100, the Mnarrow band waveforms originally transmitted at M original frequenciesmay now comprise more than M narrow band waveforms at a different rangeof frequencies. Moreover, due to transmitted OTFS waveforms impingingvarious reflectors 106, the originally transmitted signals andreflections thereof may be received at different times. As aconsequence, each receiver 104 will generally supersample or oversamplethe various received waveforms on a time-frequency grid having a finermesh than that associated with the transmitter 102. This oversamplingprocess is represented by FIGS. 17B-17D, which depict a receivertime-frequency grid having smaller time and frequency increments thanthe transmitter OTFS grid.

Each OTFS receiver 104 operates to receive the transmitted 2D Fouriertransformed information manifold over time slices having durations thatare generally less than or equal to the transmission time intervalsemployed by the transmitter 102. In one embodiment the receiver 104analyzes the received waveforms using a receiving bank of at least Mdifferent, non-overlapping, narrow-band frequency filters. The receiverwill then generally store the resoling approximation (channel convolutedversion) of the originally transmitted 2D Fourier transformedinformation manifold in receiver memory 104 m.

Once the waveforms transmitted by the transmitter 102 have beenreceived, the receiver 104 then corrects for the convolution effect ofthe channel 100 in order to facilitate recovery of an estimate of theoriginally transmitted data symbols. The receiver 104 may effect thesecorrections in a number of ways.

For example, the receiver 104 may use an inverse of the 2D symplecticFourier transform used by the transmitter 102 to transform the receivedwaveforms into an initial approximation of the information manifoldoriginally transmitted. Alternatively, the receiver 104 may first useinformation pertaining to the 2D channel state to correct thechannel-convoluted approximation of the transmitted 2D Fouriertransformed information manifold (stored in receiver memory). Followingthis correction the receiver 104 may then use the inverse of the 2Dsymplectic Fourier transform employed at the transmitter 102 to generatea received information manifold and subsequently extract estimated datasymbols.

Although the OTFS methods described herein inherently spread any givendata symbol over the entire time-frequency plane associated with atransmitter, in some embodiments it may be useful to implement anadditional spreading operation to insure that the transmitted datasymbols are uniformly distributed. This spreading operation may becarried out by the transmitter processor 102 p either prior to or afterencoding the input N×M 2D array of data symbols onto the symplecticanalysis compatible manifold. A number of spreading functions such as,for example, a 2D chirp operation, may be used for this purpose. In theevent such a spreading operation is implemented at the transmitter 102,the receiver 104 will utilize an inverse of this spreading operation inorder to decode and extract the data symbols from the various receivedinformation manifolds.

FIG. 19 illustrates transmission of a 2D Fourier transformed informationmanifold represented by an N×M structure over M frequency bands during Ntime periods of duration Tμ. In this example, each of the M frequencybands is represented by a given row and each different time period isrepresented by a given column. In the embodiment of FIG. 19 it isassumed that the OTFS transmitter is configured to transmit OTFS signalsduring without guard intervals over the allocated bandwidth, whichencompasses the M frequency bands. The bandwidth (ω₀) of each of the Mfrequency bands is, is 1/Tμ. Accordingly, if it is desired to transmitall N columns of information over a minimum time interval of N*Tμ, thenM must have a bandwidth no larger than 1/Tμ and the bandwidth used byall M filtered OTFS frequency bands cannot exceed M/T, where T is thetotal amount of time used to transmit all N columns of the 2D Fouriertransformed information manifold.

At the receiver 104, the various 2D Fourier transformed informationmanifolds may be received using banks of different, nonoverlapping,narrow-band frequency filters that are generally similar to those usedby the transmitter 102. Again, the receiver time slices and receivingbanks of filters will generally operate with finer granularity; that is,the receiver will typically operate over smaller frequency bandwidths,and shorter time slices, but over a typically broader total range offrequencies and times. Thus the receiver bin structure will preferablyoversample the corresponding transmitting time slices and transmittingbanks of different, non-overlapping, narrow-band frequency filterspreviously used by the transmitter.

As may be appreciated with reference to FIG. 19, the OTFS transmitterwill typically transmit the resulting filtered waveforms (in thisexample over all rows and successive columns), until the entire 2DFourier transformed information manifold has been transmitted. Howeverthe transmitter can either transmit the successive columns (time slices)continuously and contiguously—that is without any time gaps in-between,as more of a series of continuous longer duration waveforms, oralternatively the transmitter can put some time spacing between thevarious successive columns, thus creating a more obvious series ofwaveform bursts.

Stated differently, the transmitter can transmit the resulting filteredwaveforms as either: 1) a plurality of at least M simultaneouslytransmitted mutually orthogonal waveforms over either differentconsecutive transmitted time intervals; or 2) a plurality OTFS data orOTFS pilot bursts comprising at least M simultaneously transmittedmutually orthogonal waveform bursts over different transmitted intervalsseparated by at least one spacer time interval.

FIG. 20 shows an example of the M filtered OTFS frequency bands beingsimultaneously transmitted according to various smaller time slices Tμ.The repeating curved shapes show the center frequency for each filteredband according to g(t·e^(jkω) ⁰ ). One of the transmitted bins offrequency bandwidth, which is of size 1/T and time duration T*μ, isshown in more detail. Again, as previously discussed, in a preferredembodiment the OTFS receiver will use oversampling, and thus use finergranularity bins that nonetheless may extend over a broader range oftimes and frequencies so as to catch signals with high degrees of delayor Doppler frequency shift.

Stated differently, in some embodiments, the non-overlapping,narrow-band frequency filters used at the transmitter may be configuredto pass frequencies from the various 2D Fourier transformed Informationmanifolds that are proportional to a filter function g(t·e^(jkω) ⁰ ),where j is the square root of −1, t corresponds to a given time slice ofduration Tμ chosen from a 2D Fourier transformed information manifold,and k corresponds to a given row position in a given 2D Fouriertransformed information manifold, where k varies between 1 and M. Inthis example, the bandwidth, ω₀, in frequency units Hz, can beproportional to 1/T, and T=M/(allowed wireless bandwidth).

As may be appreciated from FIGS. 19 and 20, the various 2D Fouriertransformed information manifolds can have overall dimensions NT_(μ)according to a time axis and M/T according to a frequency axis, and each“cell” or “bin” in the various 2D Fourier transformed informationmanifold may have overall dimensions proportional to Tμ according to atime axis and 1/T according to a frequency axis.

FIG. 21 provides another example of OTFS waveforms being transmittedaccording to various smaller time slices Tμ. In the illustration of FIG.21 the amplitude or extent of modulation of the various waveforms as afunction of time is also shown.

In some embodiments it may be useful to modulate the transmittedwireless OTFS waveforms using an underlying modulation signal thatallows the receiver to distinguish where, on the original 2D time andfrequency grid, a given received signal originated. This may, forexample, assist an OTFS receiver in distinguishing the various types ofreceived signals, and in distinguishing direct signals from various timedelayed and/or frequency shifted reflected signals. In these embodimentsgrid, bin, or lattice locations of the originally transmitted OTFSwaveforms may be distinguished by determining time and frequency relatedparameters of the received waveforms. For example, in the presentlydiscussed “symplectic” implementations, where each “row” of the 2DFourier transformed information manifold is passed through a narrow bandfilter that operates according to parameters such as g(t·e^(jkω) ⁰ ),the “kω₀” term may enable the receiver to distinguish any given incomingOTFS waveform by its originating “column” location “t”. In this case thereceiver should also be able to determine the bin (grid, lattice)location of the various received waveforms by determining both the t(time related) and k (frequency related) values of the various receivedwaveforms. These values may then be used during subsequent deconvolutionof the received signals.

If further distinguishability of the bin (grid lattice) originating timeand frequency origins of the received OTFS signals is desired, then anadditional time and/or frequency varying modulation scheme may also beimposed on the OTFS signals, prior to transmission, to allow the OTFSreceiver to further distinguish the bin (grid, lattice) origin of thevarious received signals.

In alternative embodiments either the information manifold or the 2DFourier transformed information manifolds may be sampled and modulatedusing Dirac comb methods. The Dirac combs utilized by these methods maybe, for example, a periodic tempered distribution constructed from Diracdelta functions.

Attention is now directed to FIG. 22, which provides a blockdiagrammatic representation of an exemplary process 2200 of OTFStransmission and reception in accordance with the present disclosure.The process 2200 begins with the packaging of data for transmission andits optional precoding to correct for known channel impairments (stage2210). This material is then processed by a 2D Fourier Transform (suchas a symplectic Fourier transform, discrete symplectic Fouriertransform, or finite symplectic Fourier transform) (stage 2220).Following this processing the results are then passed through a filterbank (FB) and transmitted over a series of time intervals Tμ (stage2230). The transmitted wireless OTFS waveforms then pass through thecommunications or data channel (C), where they are subject to variousdistortions and signal impairments (stage 2240). At the receiver, thereceived waveforms are received according to a filter bank at varioustime intervals (stage 2250). The receiver filter bank (FB*) may be anoversampled filter bank (FB*) operating according to oversampled timedurations that may be a fraction of the original time intervals Tμ. Thisoversampling enables the received signals to be better analyzed forchannel caused time delays and frequency shifts at a high degree ofresolution. At a stage 2260 the received material is analyzed by aninverse 2D Fourier Transform (2D-FT_(s)) (which again may be an inversesymplectic Fourier transform, inverse discrete symplectic Fouriertransform, or inverse finite symplectic Fourier transform). The resultsmay then be further corrected for channel distortions using, forexample, 2D channel state information (stage 2270). In other embodimentsstage 2270 may precede stage 2260.

Further Mathematical Characterization of OTFS Modulation and Derivationof the Two-Dimensional (2D) Channel Model

In what follows we further develop the OTFS communication paradigmfocusing on the central role played by the Heisenberg representation andthe two dimensional symplectic Fourier transform. A principal technicalresult of this development is a rigorous derivation of the OTFStwo-dimensional channel model.

0. Introduction

Orthogonal time frequency space is a novel modulation scheme capable ofbeing implemented by communication transceivers that converts thedynamic one dimensional wireless medium into a static two dimensionallocal ISI channel by putting the time and frequency dimensions on anequal footing. Among the primary benefits of an OTFS transceiverrelative to a conventional transceiver are the following:

-   -   1. Fading. Elimination of fading both time and frequency        selective.    -   2. Diversity. Extraction of all diversity branches in the        channel.    -   3. Stationarity. All symbols experience the same distortion.    -   4. CSI. Perfect and efficient channel state information (CSI).

In a sense, the OTFS transceiver establishes a virtual wire through acommunication medium, thus allowing the application of conventionalwired DSP technologies in the wireless domain. Embodiments of the OTFStransceiver are based on principles from representation theory,generalizing constructions from classical Fourier theory. On theoperational level, OTFS may be roughly characterized as an applicationof the two dimensional Fourier transform to a block of filtered OFDMsymbols. OTFS is a true two dimensional time-frequency modulation, andmay incorporate both two dimensional time-frequency filtering and twodimensional equalization techniques. In what follows we provide a formalmathematical development of the OTFS transceiver, focusing on a rigorousderivation of the two dimensional channel model.

OTFS and Lattices

We first choose an undersampled time-frequency lattice, that is, a twodimensional lattice of density smaller or equal than 1. Theundersampling condition is essential for perfect reconstruction,however, it seems to limit the delay-Doppler resolution of the channelacquisition. In contrast, radar theory amounts to choosing anoversampled time frequency lattice of density greater or equal than 1where the oversampling condition is essential for maximizing thedelay-Doppler resolution of target measurement. As it turns out, thesymplectic (two dimensional) Fourier transform intertwines betweencommunication and radar lattices. The OTFS communication paradigm is tomultiplex information symbols on an oversampled high resolution radarlattice and use the symplectic Fourier transform together with twodimensional filtering to convert back to communication coordinates. Thisallows OTFS to reap the benefits of both worlds—high resolutiondelay-Doppler channel state measurement without sacrificing spectralefficiency. In particular, the OTFS channel model may be thought of as ahigh resolution delay-Doppler radar image of the wireless medium.

The Wireless Channel

In order to understand OTFS, it is beneficial to understand the wirelesschannel as a mathematical object. Let H=L²(R) denote the vector space of“physical” waveforms defined on the time domain. The physics of thewireless medium is governed by the phenomena of multipath reflection,that is, the transmitted signal is propagating through the atmosphereand reflected from various objects in the surrounding. Some of theobjects, possibly including the transmitter and the receiver, are movingat a non-zero velocity. Consequently, (under some mild “narrow band”assumption) the received signal is a superposition of time delays andDoppler shifts of the transmitted signal where the delay in time iscaused by the excess distance transversed by the reflected waveform andthe Doppler shift is caused by the relative velocity between thereflector and the transmitting and/or receiving antennas.Mathematically, this amounts to the fact that the wireless channel canbe expressed as a linear transformation C: H→H realized as a weightedsuperposition of multitude of time delays and Doppler shifts, namely:

$\begin{matrix}{{{{C(\varphi)}(t)} = {\int{\int_{\tau,v}{{h\left( {\tau,v} \right)}e^{2\pi\;{{iv}{({t - \tau})}}}{\varphi\left( {t - \tau} \right)}d\;\tau\;{dv}}}}},} & (0.1)\end{matrix}$for every transmit waveform ϕϵH. From Equation (0.1) one can see thatthe channel C is determined by the function h that depends on twovariables τ and v, referred to as delay and Doppler. The pair (τ, v) canbe viewed as a point in the plane V=R², referred to as the delay Dopplerplane. Consequently, h is a kind of a two dimensional (delay Doppler)impulse response characterizing the wireless channel. However, oneshould keep in mind that this terminology is misleading since the actionof h given by (0.1) is not a convolution action.

Fading

One basic physical phenomena characteristic to the wireless channel isfading. The phenomena of fading corresponds to local attenuation in theenergy profile of the received signal as measured over a specificdimension. It is customary to consider two kind of fadings: timeselective fading and frequency selective fading. The first is caused bydestructive superposition of Doppler shifts and the second is caused bydestructive superposition of time delays. Since the wireless channelconsists of combination of both time delays and Doppler shifts itexhibits both types of fading. Mitigating the fading phenomena is asignificant motivation behind the development of the OTFS transceiver.

The Heisenberg Representation

One key observation is that the delay Doppler channel representationgiven in Equation (0.1) is the application of a fundamental mathematicaltransform, called the Heisenberg representation, transforming betweenfunctions on the delay Doppler plane V and linear operators on thesignal space H. To see this, let us denote by L_(r) and M_(v) are theoperations of time delay by τ and Doppler shift by v respectively, thatis:L _(r)(ϕ)(t)=ϕ(t−τ),M _(v)(ϕ)(t)=e ^(2πi vt)ϕ(t),

for every ϕϵH. Using this terminology, we can rewrite channel equation(0.1) in the following form:

$\begin{matrix}\begin{matrix}{{{C(\varphi)}(t)} = {\int{\int_{\tau,v}{{h\left( {\tau,v} \right)}L_{\tau}{M_{v}(\varphi)}d\;\tau\;{dv}}}}} \\{= {\left( {\int{\int_{\tau,v}{{h\left( {\tau,v} \right)}L_{\tau}M_{v}d\;\tau\;{dv}}}} \right){(\varphi).}}}\end{matrix} & (0.2)\end{matrix}$

Let us define the Heisenberg representation to be the transform taking afunction a: V→C to the linear operator Π(a):H→H, given by:

$\begin{matrix}{{\Pi(a)} = {\int{\int_{\tau,v}{{a\left( {\tau,v} \right)}L_{\tau}M_{v}d\;\tau\;{{dv}.}}}}} & (0.3)\end{matrix}$

We refer to the function a as the delay Doppler impulse response of theoperator Π(a). Taking this perspective, we see that the wireless channelis an application of the Heisenberg representation to a specificfunction h on the delay Doppler plane. This higher level of abstractionestablishes the map Π as the fundamental object underlying wirelesscommunication. In fact, the correspondence a↔Π(a) generalizes theclassical correspondence between a stationary linear system and a onedimensional impulse response to the case of arbitrary time varyingsystems (also known as linear operators). In this regard, the mainproperty of the Heisenberg representation is that it translates betweencomposition of linear operators and an operation of twisted convolutionbetween the corresponding impulse responses. In more details, if:A=Π(a),B=Π(b),then we have:A∘B=Π(a* _(t) b),  (0.4)

where *_(t) is a non commutative twist of two dimensional convolution.Equation (0.4) is key to the derivation of the two dimensional channelmodel—the characteristic property of the OTFS transceiver.

The OTFS Transceiver and the 2D Channel Model

The OTFS transceiver provides a mathematical transformation having theeffect of converting the fading wireless channel into a stationary twodimensional convolution channel. We refer to this property as the twodimensional channel model.

Formally, the OTFS transceiver may be characterized as a pair of lineartransformations (M, D) where M is termed a modulation map and D istermed a demodulation map and is the inverse of M. According to the OTFSparadigm the information bits are encoded as a complex valued functionon V which periodic with respect to a lattice Λ^(⊥)⊂V called thereciprocal communication lattice. Note that the term “reciprocal” isused to suggest a type of duality relation between Λ^(⊥) and a moreconventional lattice Λ, called the primal communication lattice. If wedenote by C(V)_(Λ) _(⊥) the vector space of Λ^(⊥)-periodic functions onV then the OTFS modulation is a linear transformation:M:C(V)_(Λ) _(⊥) →H.  (0.5)

Geometrically, one can think of the information as a function on a twodimensional periodic domain (a donut) obtained by folding V with respectto the lattice Λ^(⊥). Respectively, the demodulation map is a lineartransformation acting in the opposite direction, namely:D:H→C(V)_(Λ) _(⊥) .  (0.6)

The precise mathematical meaning of the two dimensional channel model isthat given an information function xϵC(V)_(Λ) _(⊥) , we have:D∘C∘M(x)=c*x,  (0.7)

where * stands for periodic convolution on the torus and the function cis a periodization with respect to the reciprocal lattice Λ^(⊥) of thedelay Doppler impulse response h of the wireless channel, that is:c=per_(Λ) _(⊥) (h).  (0.8)

Equations (0.7) and (0.8) encodes the precise manner of interactionbetween the OTFS transceiver and the wireless channel.

The remainder of this explanation of OTFS method and the OTFStransceiver is organized as follows:

Section 1 discusses several basic mathematical structures associatedwith the delay Doppler plane V. We begin by introducing the symplecticform on V which is an antisymmetric variant of the more familiarEuclidean form used in classical signal processing. We than discusslattices which are two dimensional discrete subdomains of V. We focusour attention to the construction of the reciprocal lattice. Thereciprocal lattice plays a pivotal role in the definition of the OTFStransceiver. We than proceed to discuss the dual object of a lattice,called a torus, which is a two dimensional periodic domain obtained byfolding the plain with respect to a lattice.

Section 2 discusses the symplectic Fourier transform, which is a variantof the two dimensional Fourier transform defined in terms of thesymplectic form on V. We discuss three variants of the symplecticFourier transform: the continuous, the discrete and the finite. Weexplain the relationships between these variants.

Section 3 discusses the Heisenberg representation and its inverse—theWigner transform. In a nutshell, the Heisenberg representation is thestructure that encodes the precise algebraic relations between theoperations of time delay and Doppler shifts. We relate the Wignertransform to the more familiar notions of the ambiguity function and thecross ambiguity function. We conclude with a formulation of thefundamental channel equation.

Section 4 discusses the continuous variant of the OTFS transceiver. Webegin by specifying the parameters defining the OTFS transceiver. Thenwe proceed to define the modulation and demodulation maps. We concludethe section with a derivation of the two dimensional channel model fromfirst principles.

Section 5 discusses the finite variant of the OTFS transceiver. In anutshell, the finite variant is obtained from the continuous variant bysampling the reciprocal torus along finite uniformly distributedsubtorus. We define the finite OTFS modulation and demodulation maps. Wethen formulate the finite version of the two dimensional channel model,explaining the finite two dimensional impulse response is therestriction of the continuous one to the finite subtorus. We concludethis section with an explicit interpretation of the modulation formulain terms of classical DSP operations.

1. The Delay-Doppler Plane

1.1 The Symplectic Plane

The delay Doppler plane is a two dimensional vector space over the realnumbers. Concretely, we take V=R² where the first coordinate is delay,denoted by τ and the second coordinate is Doppler, denoted by v. Thedelay Doppler plane is equipped with an intrinsic geometric structureencoded by a symplectic form (also called symplectic inner product orsymplectic pairing). The symplectic form is a pairing ω:V×V→R defined bythe determinant formula:

$\begin{matrix}{{{\omega\left( {v^{\prime},v} \right)} = {{- {\det\begin{bmatrix}\tau & \tau^{\prime} \\v & v^{\prime}\end{bmatrix}}} = {{v\;\tau^{\prime}} - {\tau\; v^{\prime}}}}},} & (1.1)\end{matrix}$

where v=(τ, v) and v′=(τ′, v′). Note that the symplectic form, incontrast to its Euclidean counterpart, is anti-symmetric, namely ω(v,v′)=−ω(v′, v) for every v, v′ϵV. Consequently, the symplectic product ofa vector with itself is always equal zero, that is ω(v, v)=0, for everyvϵV. As it turns out, the fabric of time and frequency is governed by asymplectic structure.

1.1.1 Functions on the Plane.

We denote by C(V) the vector space of complex valued functions on V. Wedenote by * the operation of linear convolution of functions on V. Givena pair of functions ƒ,gϵ C(V), their convolution is defined by:

$\begin{matrix}\begin{matrix}{{f*{g(v)}} = {\int_{{v_{1} + v_{2}} = v}{{f\left( v_{1} \right)}{g\left( v_{2} \right)}}}} \\{{= {\int_{v^{\prime} \in V}{{f\left( v^{\prime} \right)}{g\left( {v - v^{\prime}} \right)}d\; v^{\prime}}}},}\end{matrix} & (1.2)\end{matrix}$

for every vϵV.

1.2 Lattices

A lattice Λ⊂V is a commutative subgroup isomorphic to Z² defined asfollows:

$\begin{matrix}{\Lambda = {{Zv}_{i} \oplus {Zv}_{2}}} \\{{= \left\{ {{{av}_{1} + {{bv}_{2}\text{:}a}},{b \in Z}} \right\}},}\end{matrix}$

where v₁, v₂ϵV are linear independent vectors. In words, Λ consists ofall integral linear combinations of the vectors v₁ and v₂. See FIG. 23.The vectors v₁ and v₂ are called generators of the lattice. The volumeof Λ is, by definition, the volume of a fundamental domain. One can showthat:vol(Λ)=|ω(v ₁ ,v ₂)|.  (1.3)

when vol(Λ)≥1 the lattice is called undersampled and when vol(Λ)≤1 thelattice is called oversampled. Finally, in case vol(Λ)=1 the lattice iscalled critically sampled.

Example 1.1 (Standard Communication Lattice)

Fix parameters T≥0 and μ≥1. Let:

$\begin{matrix}\begin{matrix}{\Lambda_{T,\mu} = {{{ZT}\;\mu} \oplus {Z\; 1\text{/}T}}} \\{= {\left\{ {{\left( {{{K \cdot T}\;\mu} + {{L \cdot 1}\text{/}T}} \right)\text{:}K},{L \in Z}} \right\}.}}\end{matrix} & (1.4)\end{matrix}$

We have that vol(Λ_(T,μ))=μ. We refer to Λ_(T/μ) as the standardcommunication lattice.

1.2.1 Reciprocal Lattice.

Given a lattice Λ⊂V, its orthogonal complement lattice is defined by:Λ^(⊥) ={vϵV:ω(v,λ)ϵZforeveryλϵΛ}.  (1.5)

In words, Λ^(⊥) consists of all vectors in V such that their symplecticpairing with every vector in Λ is integral. One can show that Λ^(⊥) isindeed a lattice. We refer to Λ^(⊥) as the reciprocal lattice of Λ. Onecan show that:vol(Λ^(⊥))=1/vol(Λ),  (1.6)which implies that Λ is undersampled if and only if Λ^(⊥) isoversampled. This means that reciprocity interchanges between coarse(undersampled) lattices and fine (oversampled) lattices. Anotherattribute concerns how lattice inclusion behaves under reciprocity.Given a pair consisting of a lattice Λ⊂V and a sublattice Λ₀⊂Λ, one canshow that the the inclusion between the reciprocals is reversed, thatis:Λ^(⊥)⊂Λ₀ ^(⊥).  (1.7)

Example 1.2

Consider the standard communication lattice Λ_(T,μ). Its reciprocal isgiven by:(Λ_(T,μ))^(⊥) =ZT⊕Z1/Tμ.  (1.8)

See FIGS. 24A and 24B, which respectively illustrate a standardcommunication lattice and the reciprocal of the standard communicationlattice. Indeed, we have that:

${\omega\left( {\begin{bmatrix}{KT} \\{L\text{/}T\;\mu}\end{bmatrix},\begin{bmatrix}{K^{\prime}T\;\mu} \\{L^{\prime}\text{/}T}\end{bmatrix}} \right)} = {{{LK}^{\prime} - {KL}^{\prime}} \in {Z.}}$

Note that vol(Λ_(T,μ))^(⊥)=1/μ which means that as the primal latticebecomes sparser, the reciprocal lattice becomes denser.

1.2.2 Functions on a Lattice.

We denote by C(Λ) the vector space of complex valued functions on thelattice. We denote by R^(A):C(V)→C(Λ) the canonical restriction map,given by:R ^(Λ)(ƒ)(λ)=ƒ(λ),for every ƒϵC(V) and λϵΛ. We denote by * the convolution operationsbetween functions on Λ. Given ƒ,gϵC(Λ), their convolutions is definedby:

$\begin{matrix}\begin{matrix}{{f*{g(\lambda)}} = {\sum\limits_{{\lambda_{1} + \lambda_{2}} = \lambda}{{f\left( \lambda_{1} \right)}{g\left( \lambda_{2} \right)}}}} \\{{= {\sum\limits_{\lambda^{\prime} \in \Lambda}{f\left( \lambda^{\prime} \right){g\left( {\lambda - \lambda^{\prime}} \right)}}}},}\end{matrix} & (1.9)\end{matrix}$for every λϵΛ.

1.3 Tori

A torus Z is a two dimensional periodic domain that constitutes thegeometric dual of a lattice Λ. Formally, Z is the continuous groupobtained as the quotient of the vector space V by the lattice Λ, namely:Z=V/Λ.  (1.10)

In particular, a point zϵZ is by definition a Λ-coset in V, namely:z=v+Λ,  (1.11)for some vϵV. An alternative, albeit less canonical, way to construct Zis to glue opposite faces of a fundamental domain of Λ. Geometrically, Zhas the shape of a “donut” obtained by folding the plane V with respectto the lattice Λ. We refer to Z as the torus associated with Λ orsometimes also as the dual of Λ. Note that a torus is the twodimensional counterpart of a circle, where the second is obtained byfolding the line R with respect to a one dimensional lattice ZT⊂R.

Example 1.3 (Standard Communication Torus)

As shown in FIG. 25, the torus associated with the standardcommunication lattice Λ_(T,μ) is given by:

$\begin{matrix}\begin{matrix}{Z_{T,\mu} = {V\text{/}\Lambda_{T,\mu}}} \\{{= {{R\text{/}{ZT}\;\mu} \oplus {R\text{/}Z\; 1\text{/}T}}};} \\{\left. {{\left\lbrack {0,{T\;\mu}} \right) \times 0},{1\text{/}T}} \right).}\end{matrix} & (1.12)\end{matrix}$Geometrically, Z_(T,μ) is the Cartesian product of two circles; one ofdiameter Tμ and the other of diameter 1/T. We refer to Z_(T,μ) as thestandard communication torus.

1.3.1 Functions on Tori.

We denote by C(Z) the vector space of complex valued functions on atorus Z=V/Λ. A function on Z is naturally equivalent to a function ƒ:V→Cperiodic with respect to translations by elements of the lattice Λ, thatis:ƒ(v+λ)=ƒ(v),  (1.13)for every vϵV and λϵΛ. Hence, the vector space of functions on Zcoincides with the subspace of Λ periodic functions on V, that is,C(Z)=C(V)_(Λ). Consequently, we have a natural periodization mapR_(Λ):C(V)→C(Z), given by:

$\begin{matrix}{{{{R_{\Lambda}(f)}(v)} = {\sum\limits_{\lambda \in \Lambda}{f\left( {v + \lambda} \right)}}},} & (1.14)\end{matrix}$for every ƒϵC(V) and vϵV. We denote by * the operation of cyclicconvolution of functions on Z. Given a pair of functions ƒ,gϵ C(Z),their convolution is defined by:

$\begin{matrix}\begin{matrix}{{f*{g(v)}} = {\int_{{v_{1} + v_{2}} = v}{{f\left( v_{1} \right)}{g\left( v_{2} \right)}}}} \\{{= {\int_{v^{\prime} \in Z}{{f\left( v^{\prime} \right)}{g\left( {v - v^{\prime}} \right)}{dv}^{\prime}}}},}\end{matrix} & (1.15)\end{matrix}$for every vϵV. Note that integration over the torus Z amounts tointegration over a fundamental domain of the lattice Λ.

1.4 Finite Tori

A finite torus Z₀ is a domain associated with a pair consisting of alattice Λ⊂V and a sublattice Λ₀⊂A. Formally, Z₀ is the finite groupdefined by the quotient of the lattice Λ by the sublattice Λ₀, that is:Z ₀=Λ/Λ₀.  (1.16)

In particular, a point zϵZ₀ is a Λ₀-coset in Λ, namely:z=λ+Λ ₀,  (1.17)for some λϵΛ. Geometrically, Z₀ is a finite uniform sampling of thecontinuous torus Z=V/Λ₀ as we have a natural inclusion:Λ/Λ₀ °V/Λ ₀.  (1.18)

Example 1.4 (the Standard Communication Finite Torus)

Consider the standard communication lattice Λ_(T,μ). Fix positiveintegers n, mϵN^(≥1). Let

(Λ_(T, μ))_(n, m)be the sublattice defined by:

$\begin{matrix}{\left( \Lambda_{T,\mu} \right)_{n,m} = {{{ZnT}\;\mu} \oplus {{Zm}\text{/}{T.}}}} & (1.19)\end{matrix}$

The finite torus associated with

(Λ_(T, μ))_(n, m) ⋐ Λ_(T, μ)is given by (see FIG. 26):

$\begin{matrix}\begin{matrix}{Z_{T,\mu}^{m,n} = {\Lambda_{T,\mu}\text{/}\left( \Lambda_{T,\mu} \right)_{n,m}}} \\{{= {{ZT}\;\mu\text{/}{ZnT}\;\mu \times Z\; 1\text{/}T\text{/}{Zm}\text{/}T}};} \\{Z\text{/}{nZ} \times Z\text{/}{{mZ}.}}\end{matrix} & (1.20)\end{matrix}$Concluding, the finite torus Z_(T,μ) ^(m,n) is isomorphic to theCartesian product of two cyclic groups; one of order n and the other oforder m. We refer to Z_(T,μ) ^(m,n) as the standard communication finitetorus.

1.4.1 Functions on Finite Tori.

We denote by C(Z₀) the vector space of complex valued functions on afinite torus Z₀=Λ/Λ₀. A function on Z₀ is naturally equivalent to afunction ƒ:Λ→C that is periodic with respect to translations by thesublattice Λ₀, that is:ƒ(λ+λ₀)=ƒ(λ),  (1.21)for every λϵΛ and λ₀ϵΛ₀. Hence, the vector space C(Z₀) coincides withthe subspace of Λ₀ periodic functions on Λ, that is, C(Z₀)=C(Λ)_(Λ) ₀ .Consequently, we have a natural periodization map R_(Λ) ₀ :C(Λ)→C(Z₀)given by:

$\begin{matrix}{{{{R_{\Lambda_{0}}(f)}(\lambda)} = {\sum\limits_{\lambda_{0} \in \Lambda_{0}}{f\left( {\lambda + \lambda_{0}} \right)}}},} & (1.22)\end{matrix}$for every ƒϵC(A) and λϵΛ. We denote by * the operation of finite cyclicconvolution of functions on Z₀. Given a pair of functions ƒ,gϵ C(Z₀),their convolution is defined by:

$\begin{matrix}\begin{matrix}{{f*{g(\lambda)}} = {\sum\limits_{{\lambda_{1} + \lambda_{2}} = \lambda}{{f\left( \lambda_{1} \right)}{g\left( \lambda_{2} \right)}}}} \\{= {\sum\limits_{\lambda^{\prime} \in Z}{{f\left( \lambda^{\prime} \right)}{g\left( {\lambda - \lambda^{\prime}} \right)}}}}\end{matrix} & (1.23)\end{matrix}$for every vϵV. Note that summation over the finite torus Z₀ amounts tosummation over a fundamental domain of the sublattice Λ₀ in thesuperlattice Λ.

1.4.2 Reciprocity Between Finite Tori.

Given a finite torus Z₀=Λ/Λ₀, we denote by Z^(⊥) the finite torusassociated with the reciprocal pair Λ^(⊥)⊂₀ ^(⊥), that is:Z ₀ ^(⊥)=Λ₀ ^(⊥)/Λ^(⊥).  (1.24)

We refer to Z₀ ^(⊥) as the reciprocal finite torus. Although differentas sets, one can show that, in fact, Z₀ and Z₀ ^(⊥) are isomorphic asfinite groups.

Example 1.5

Consider the pair consisting of the standard communication latticeΛ_(T,μ) and the sublattice

(Λ_(T, μ))_(m, n) ⋐ Λ_(T, μ).As shown above, the finite torus associated with

(Λ_(T, μ))_(n, m) ⋐ Λ_(T, μ)is isomorphic to:Z ₀ ;Z/Zn×Z/Zm.

The reciprocal lattices are given by:(Λ_(T,μ))^(⊥) =ZT⊕Z1/Tμ,(Λ_(T,μ))_(m,n) ^(⊥) =ZT/m⊕Z1/nTμ.

Consequently, the reciprocal finite torus is given by:

$\begin{matrix}{Z_{0}^{\bot} = {\left( \Lambda_{T,\mu} \right)_{m,n}^{\bot}\text{/}\left( \Lambda_{T,\mu} \right)^{\bot}}} \\{{= {Z\left( {T\text{/}m} \right)\text{/}{ZT} \times {Z\left( {1\text{/}{nT}\;\mu} \right)}\text{/}{Z\left( {1\text{/}T\;\mu} \right)}}};} \\{Z\text{/}{mZ} \times Z\text{/}{{nZ}.}}\end{matrix}$We see that Z₀ and Z₀ ^(⊥) are isomorphic as finite groups as bothgroups are isomorphic to the Cartesian product (albeit in differentorder) of two cyclic groups, one of order n and the other of order m.2 The Symplectic Fourier Transform

In this section we introduce a variant of the two dimensional Fouriertransform, associated with the symplectic form, called the symplecticFourier transform. Let ψ:R→C^(x) denote the standard complex exponentialfunction:ψ(z)=e ^(2πiz),  (2.1)

for every z⊂R.

2.1 Properties of the Symplectic Fourier Transform

The symplectic Fourier transform is a variant of the two dimensionalFourier transform that is associated with the symplectic form ω.Formally, the symplectic Fourier transform is the linear transformationSF:C(V)→C(v) defined by the rule:

$\begin{matrix}\begin{matrix}{{{{SF}(f)}(u)} = {\int_{v \in V}{{\psi\left( {- {\omega\left( {u,v} \right)}} \right)}{f(v)}{dv}}}} \\{{= {\int_{\tau,{v \in R}}{{\psi\left( {{tv} - {f\;\tau}} \right)}{f\left( {\tau,v} \right)}d\;\tau\;{dv}}}},}\end{matrix} & (2.2)\end{matrix}$

for every ƒϵC(V) and u=(t,f). We refer to the coordinates (t, f) of thetransformed domain as time and frequency, respectively.

In general, the inverse transform of (2.2) is given by the formula:

$\begin{matrix}\begin{matrix}{{{{SF}^{- 1}(f)}(v)} = {\int_{u \in V}{{\psi\left( {+ {\omega\left( {u,v} \right)}} \right)}{f(u)}{du}}}} \\{= {\int_{t,{f \in R}}{{\psi\left( {{\tau\; f} - {vt}} \right)}{f\left( {t,f} \right)}{dtdf}}}}\end{matrix} & (2.3)\end{matrix}$

However, since ω is anti-symmetric, we have that SF⁻¹=SF. Namely, thesymplectic Fourier transform is equal to its inverse.

2.1.1 Interchanging Property.

The symplectic Fourier transform interchanges between functionmultiplication and function convolution as formulated in the followingproposition.

Proposition 2.1 (Interchanging property). The following conditions hold:SF(ƒ·g)=SF(ƒ)*SF(g),SF(ƒ*g)=SF(ƒ)·SF(g),  (2.4)

for every ƒ,gϵC(V).

In fact, the interchanging property follows from a more fundamentalproperty that concerns the operations of two dimensional translation andsymplectic modulation.

Translation: given a vector v₀ϵV, define translation by v₀ to be thelinear transformation L_(v) ₀ :C(V)→C(V), given by:L _(v) ₀ (ƒ)(v)=ƒ(v−v ₀),  (2.5)

for every ƒϵC(V).

Modulation: given a vector v₀ϵV, define symplectic modulation by v₀ tobe the linear transformation M_(v) ₀ :C(V)→C(V), given by:M _(v) ₀ (ƒ)(v)=ψ(ω(v ₀ ,v))ƒ(v),  (2.6)

for every ƒϵC(V).

Perhaps the most fundamental property of the symplectic Fouriertransform is that it interchanges between translation and symplecticmodulation. This property is formulated in the following proposition.

Proposition 2.2 (interchanging translation with symplectic modulation).The following conditions hold:SF∘L _(v) ₀ =M _(v) ₀ ∘SF,SF∘M _(v) ₀ =L _(v) ₀ ∘SF,

for every v₀ϵV.

2.2 The Discrete Symplectic Fourier Transform

The discrete symplectic Fourier transform relates between functions oftwo discrete variables and functions of two continuous periodicvariables. The formal definition assumes a choice of a lattice Λ⊂V. LetΛ^(⊥)⊂V be the reciprocal lattice and let Z^(⊥) denote the torusassociated with Λ^(⊥), that is:Z ^(⊥) =V/Λ ^(⊥).

We refer to Z^(⊥) as the reciprocal torus. The discrete symplecticFourier transform is the linear transformation SF_(Λ):C(Λ)→C(Z^(⊥))given by:

$\begin{matrix}{{{{{SF}_{\Lambda}(f)}(u)} = {c \cdot {\sum\limits_{\lambda \in \Lambda}{{\psi\left( {- {\omega\left( {u,\lambda} \right)}} \right)}{f(\lambda)}}}}},} & (2.7)\end{matrix}$for every ƒϵC(Λ) and uϵV where c is a normalization coefficient taken tobe c=vol(Λ). Note, that fixing the value of λϵΛ, the function ψ(−ω(u,λ))ƒ(Λ) is periodic with respect to the reciprocal lattice hence is afunction on the reciprocal torus. The inverse transform SF_(Λ)⁻¹:C(Z^(⊥))→C(Λ) is given by:

$\begin{matrix}{{{{{SF}_{\Lambda}^{- 1}(f)}(\lambda)} = {\int_{\square}{{\psi\left( {- {\omega\left( {\lambda,u} \right)}} \right)}{f(u)}{du}}}},} & (2.8)\end{matrix}$for every ƒϵC(Λ). Note that taking the integral over the torus Z^(⊥) isequivalent to integrating over a fundamental domain of the latticeΛ^(⊥).

2.2.1 Discrete Interchanging Property.

The discrete symplectic Fourier transform interchanges between functionmultiplication and function convolution as formulated in the followingproposition.

Proposition 2.3 (Discrete interchanging property). The followingconditions hold:

$\begin{matrix}{{{{SF}_{\Lambda}\left( {f \cdot g} \right)} = {{{SF}_{\Lambda}(f)}*{{SF}_{\Lambda}(g)}}},} & (2.9) \\{{{\frac{1}{\sqrt{c}}{{SF}_{\Lambda}\left( {f*g} \right)}} = {\frac{1}{\sqrt{c}}{{{SF}_{\Lambda}(f)} \cdot \frac{1}{\sqrt{c}}}{{SF}_{\Lambda}(g)}}},} & (2.10)\end{matrix}$

for every ƒ,gϵC(Λ) where * stands for periodic convolution.

2.2.2 Compatibility with the Continuous Transform.

The continuous and discrete symplectic Fourier transforms arecompatible. The compatibility relation is formulated in the followingTheorem.

Theorem 2.4 (discrete-continuous compatibility relation). We have:SF _(Λ) ∘R ^(Λ) =R _(Λ) _(⊥) ∘SF,  (2.11)SF _(Λ) ⁻¹ ∘R _(Λ) _(⊥) =R ^(Λ) ∘SF ⁻¹.  (2.12)

Stated differently, Equation (2.11) provides that taking the continuousFourier transform of a function ƒ and than periodizing with respect totranslations by the reciprocal lattice Λ^(⊥) is the same as firstrestricting ƒ to the lattice Λ and then taking the discrete Fouriertransform.

2.3 The Finite Symplectic Fourier Transform

The finite symplectic Fourier transform relates functions of two finiteperiodic variables. The formal definition assumes a pair consisting of alattice Λ⊂V and a sublattice Λ₀⊂Λ. We denote by Z₀ the finite torusassociated with this pair, that is:Z ₀=Λ/Λ₀.

Let Λ^(⊥) and Λ₀ ^(⊥) be the corresponding reciprocal lattices. Wedenote by Z^(⊥) the finite torus associated with the reciprocal pair,that is:Z ₀ ^(⊥)=Λ₀ ^(⊥)/Λ^(⊥).

The finite symplectic Fourier transform is the linear transformationSF_(Z) ₀ :C(Z₀)→C(Z₀ ^(⊥)) defined by the rule:SF _(Z) ₀ (ƒ)(μ)=c· _(λϵZ) ₀ ψ(−ω(μ,λ))ƒ(λ),  (2.13)for every ƒϵC(Z₀) and μϵΛ₀ ^(⊥) where c is a normalization coefficienttaken to be c=vol(Λ). The inverse transform SF_(Z) ₀ ⁻¹:C(Z₀ ^(⊥))→C(Z₀)is given by:

$\begin{matrix}{{{{{SF}_{Z_{0}}^{- 1}(f)}(\lambda)} = {{\frac{1}{c_{0}} \cdot_{\mu \in Z_{0}^{\bot}}{\psi\left( {- {\omega\left( {\lambda,\mu} \right)}} \right)}}{f(\mu)}}},} & (2.14)\end{matrix}$

for every ƒϵC(Z₀ ^(⊥)) and λϵΛ where c₀ is a normalization coefficienttaken to be c₀=vol(Λ₀).

2.3.1 Finite Interchanging Property.

The finite symplectic Fourier transform interchanges between functionmultiplication and function cyclic convolution as formulated in thefollowing proposition.

Proposition 2.5 (discrete interchanging property). The followingconditions hold:

$\begin{matrix}{{{\frac{c}{c_{0}}{{SF}_{Z_{0}}\left( {f \cdot g} \right)}} = {\frac{c}{c_{0}}{{SF}_{Z_{0}}(f)}*\frac{c}{c_{0}}{{SF}_{Z_{0}}(g)}}},} & (2.15) \\{{{\frac{1}{c}{{SF}_{Z_{0}}\left( {f*g} \right)}} = {\frac{1}{c}{{{SF}_{Z_{0}}(f)} \cdot \frac{1}{c}}{{SF}_{Z_{0}}(g)}}},} & (2.16)\end{matrix}$

for every ƒ,gϵC(Z₀) where * stands for finite cyclic convolution.

Note that the normalization coefficient c/c₀ in equation (2.15) is equalthe number of points in the finite torus Z₀.

2.3.2 Compatibility with the Discrete Transform.

The discrete and finite symplectic Fourier transforms are compatible.The compatibility relation is formulated in the following Theorem.

Theorem 2.6. We have:SF _(Z) ₀ ∘R _(Λ) ₀ =R ^(Λ) ⁰ ^(⊥) ∘F _(Λ),  (2.17)SF _(Z) ₀ ⁻¹ ∘R ^(Λ) ⁰ ^(⊥) =R _(Λ) ₀ ∘SF _(Λ) ⁻¹  (2.18)

In plain language, Equation (2.17) states that taking the discretesymplectic Fourier transform of a function ƒ on a lattice Λ and thanrestricting to the reciprocal lattice Λ₀ ^(⊥) is the same as firstperiodizing ƒ with respect to translations by the sublattice Λ₀ and thantaking the finite Fourier transform.

Example 2.7

Consider the standard communication lattice Λ_(T,μ) and the sublattice

(Λ_(T, μ))_(n, m).We have the following isomorphisms:Z ₀ ;Z/nZ×Z/mZ,Z ₀ ^(⊥) ;Z/mZ×Z/nZ.

In terms of these realizations the finite symplectic Fourier transformand its inverse take the following concrete forms:

$\begin{matrix}{{{{{SF}_{Z_{0}}(f)}\left( {k,l} \right)} = {\mu_{K = 0_{L = 0}}^{n - 1^{m - 1}}{\psi\left( {{kL} - {lK}} \right)}{f\left( {K,L} \right)}}},} & (2.19) \\{{{{{SF}_{Z_{0}}^{- 1}(f)}\left( {K,L} \right)} = {{\frac{1}{{mn}\;\mu}\;}_{k = 0_{l = 0}}^{m - 1^{n - 1}}{\psi\left( {{Kl} - {Lk}} \right)}{f\left( {k,l} \right)}}},} & (2.20)\end{matrix}$

where in the first equation kϵ[0,m−1], lϵ[0,n−1] and in the secondequation Kϵ[0,n−1], Lϵ[0,m−1]. Note the minus sign in the Fourierexponent due to the symplectic pairing.

3 Heisenberg Theory

Let H denote the Hilbert space of square integrable complex functions onthe real line R. We denote the parameter of the line by t and refer toit as time. The inner product on H is given by the standard formula:

ƒ,g

= _(xϵR)

g(x)dx,  (3.1)

We refer to H as the signal space and to functions in the signal spaceas waveforms. Heisenberg theory concerns the mathematical structuresunderlying the intricate interaction between the time and frequencydimensions. In a nutshell, the theory study the algebraic relationsbetween two basic operations on functions: time delay and Doppler shift.

3.1 Time Delay and Doppler Shift

The operations of time delay and Doppler shift establish two oneparametric families of Unitary transformations on H.

3.1.1 Time Delay.

Given a real parameter τϵR the operation of time delay by τ is a lineartransformation L_(τ):H→H given byL _(τ)(ƒ)(t)=ƒ(tϵτ),  (3.2)for every ƒϵH and tϵR. One can show that L_(τ) is a Unitarytransformation, namely it preserves the inner product:

L _(τ) ƒ,L _(τ) g

=

ƒ,g

,

for every ƒ,gϵH. More over, the family of transformation {L_(τ):τϵR}satisfies:L _(τ) ₁ _(+τ) ₂ =L _(τ) ₁ ∘L _(τ) ₂ ,for every τ₁, τ₂ϵR. In particular, the operations of time delay commutewith one another, that is, L_(τ) ₁ ∘L_(τ) ₂ =L_(τ) ₂ ∘L_(τ) ₁ .

3.1.2 Doppler Shift.

Given a real parameter vϵR the operation of Doppler shift by v is alinear transformation M_(v):H→H given byM _(v)(ƒ)(t)=ψ(vt)ƒ(t),  (3.3)for every ƒϵH and tϵR. Recall that ψ stands for the standard complexexponential function ψ(z)=e^(2πiz). One can show that M_(v) is a Unitarytransformation, namely it preserves the inner product:

M _(v) ƒ,M _(v) g

=

ƒ,g

,for every ƒ,gϵH. More over, the family of transformation {M_(v):vϵR}satisfies:M _(v) ₁ _(+v) ₂ =M _(v) ₁ ∘M _(v) ₂ ,

for every v₁, v₂ϵR. In particular, the operations of time delay commutewith one another, that is, M_(v) ₁ ∘M_(v) ₂ =M_(v) ₂ ∘M_(v) ₁ .

3.2 The Heisenberg Representation

The Heisenberg representation is a mathematical structure unifying thetwo operations of time delay and Doppler shift. The main difficulty isthat these operations do not commute with one another, instead theysatisfy the following condition:L _(τ) M _(v)=ψ(−τv)M _(v) L _(τ).  (3.4)

The starting point is to consider the unified delay-Doppler lineartransformation:π(τ,v)=L _(τ) M _(v),  (3.5)for every pair of real parameters τ,vϵR. In this representation, theordered pair (τ, v) is considered as a point in the delay Doppler planeV. One can show that π(τ, v) is a Unitary transformation as compositionof such. The two dimensional family of transformations {π(v):vϵV}defines a linear transformation Π:C(V)→Hom(H,H), given by:Π(ƒ)=∫_(vϵV)ƒ(v)π(v)dv,  (3.6)for every ƒϵC(V), where the range of Π is the vector space of lineartransformations from H to itself which we denote by Hom(H,H). In words,the map Π takes a function on the delay Doppler plane and send it to thelinear transformation given by weighted superposition of delay-Dopplertransformations where the weights are specified by the values of thefunction. The map Π is called the Heisenberg representation. Afundamental fact which we will not prove is that the map Π isessentially an isomorphism of vector spaces. Hence it admits an inverseΠ⁻¹:Hom(H,H)→C(V) called the Wigner transform. The Wigner transform isgiven by:Π⁻¹(A)(v)=Tr(π(v)^(H) A).  (3.7)for every AϵHom(H,H) and vϵV. The Heisenberg representation and theWigner transform should be thought of as a “change of coordinates”converting between functions on the delay Doppler plane and lineartransformations on the signal space (which may be represented usingmatrices). To summarize, a linear transformation AϵHom(H,H) admits aunique expansion as a superposition of delay-Doppler transformations.The coefficients in this expansion are given by the function a=Π⁻¹(A).The function a is refereed to as the delay-Doppler impulse response ofthe transformation A. The Heisenberg formalism generalizes the classicalframework of time invariant linear systems to time varying linearsystems. Note that in the former, a time invariant linear transformationadmits a unique expansion as a superposition of time delays and thecoefficients in the expansion constitute the classical impulse response.

3.2.1 Ambiguity Function.

The formula of the Wigner transform of a general linear transformation,Equation (3.7), is quite abstract. Fortunately, for specific type oflinear transformations the Wigner transform takes a more explicit form.Say we are given a waveform gϵH, of unit norm ∥g∥=1. Let P_(g) denotethe orthogonal projection on the one dimensional subspace spanned by g,given by:P _(g)(ϕ)=g

g,ϕ

,  (3.8)

for every ϕϵH.

Proposition. The Wigner transform of P_(g) admits the following formula:Π⁻¹(P _(g))(v)=

π(v)g,g

,  (3.9)

for every vϵV.

Denote A_(g)=Π⁻¹(P_(g)) and refer to this function as the ambiguityfunction of g. We have:Π(A _(g))=P _(g).  (3.10)

The above equation means that A_(g) is the coefficients in thedelay-Doppler expansion of the operator P_(g)—this is the Heisenberginterpretation of the ambiguity function.

3.2.2 Cross Ambiguity Function.

The cross ambiguity function is a generalization of the ambiguityfunction to the case of two waveforms g₁,g₂ϵH where g₁ is assumed to beof unit norm. Let P_(g) ₁ _(,g) ₂ denote the following rank one lineartransformation on H:P _(g) ₁ _(,g) ₂ (ϕ)=g ₂

g ₁,ϕ

,  (3.11)

for every ϕϵH.

Proposition. The Wigner transform of P_(g) ₁ _(,g) ₂ admits thefollowing formula:Π⁻¹(P _(g) ₁ _(,g) ₂ )(v)=

π(v)g ₁ ,g ₂

,  (3.12)

for every vϵV.

Denote A_(g) ₁ _(,g) ₂ =Π⁻¹(P_(g) ₁ _(,g) ₂ ) and refer to this functionas the cross ambiguity function of g₁ and g₂. We have:Π(A _(g) ₁ _(,g) ₂ )=P _(g) ₁ _(,g) ₂ .  (3.13)Hence, according to the Heisenberg interpretation, the cross-ambiguityfunction is the coefficients in the delay-Doppler expansion of theoperator P_(g) ₁ _(,g) ₂ .

3.3 Heisenberg Interchanging Property

The main property of the Heisenberg representation is that itinterchanges between the operation of composition of lineartransformations on H and a twisted version of the convolution operationof functions on V. In order to define the operation of twistedconvolution we consider the form β:V×V→V, given by:β(v,v′)=vτ′,  (3.14)

where v=(τ, v) and v′=(τ′, v′). The form β satisfies the “polarization”condition:β(v,v′)−β(v′,v)=ω(v,v′),  (3.15)

for every v, v′ϵV. Given a pair of functions ƒ,gϵC(V) their twistedconvolution is defined by the following rule:

$\begin{matrix}\begin{matrix}{{f*_{t}{g(v)}} = {\int_{{{v\; 1} + {v\; 2}} = v}{{\psi\left( {\beta\left( {v_{1},v_{2}} \right)} \right)}{f\left( v_{1} \right)}{g\left( v_{2} \right)}}}} \\{= {\int_{v^{\prime} \in V}{{\psi\left( {\beta\left( {v^{\prime},{v - v^{\prime}}} \right)} \right)}{f(v)}{g\left( {v - v^{\prime}} \right)}{dv}^{\prime}}}}\end{matrix} & (3.16)\end{matrix}$

One can see that the twisted convolution operation differs from theusual convolution operation, Equation (1.2), by the multiplicativefactor ψ(β(v₁, v₂)). As a consequence of this factor, twistedconvolution is a non-commutative operation in contrast with conventionalconvolution. This non-commutativity is intrinsic to the fabric of timeand frequency. The Heisenberg interchanging property is formulated inthe following Theorem.

Theorem 3.1 (Heisenberg Interchanging property). We have:Π(ƒ* _(t) g)=Π(ƒ)∘Π(g),  (3.17)

for every ƒ,gϵC(V).

The following example is key to understanding the motivation behind theconstructions presented in this section. In a nutshell, it explains whythe twist in Formula (3.16) accounts for the phase in the commutationrelation between the time delay and Doppler shift operations, seeEquation (3.4).

Example 3.2

We verify Equation (3.17) in a concrete case. Let v=(τ, v) and v′=(τ′,v′). Consider the delta functions δ_(v) and δ_(v′). On the one hand, wehave:Π(δ_(v))=L _(τ) M _(v),Π(δ_(v′))=L _(τ′) M _(v′),

and consequently:

$\begin{matrix}\begin{matrix}{{{\Pi\left( \delta_{v} \right)} \circ {\Pi\left( \delta_{v^{\prime}} \right)}} = {L_{\tau}M_{v}L_{\tau^{\prime}}M_{v^{\prime}}}} \\{= {{\psi\left( {v\;\tau^{\prime}} \right)}L_{t}L_{\tau^{\prime}}M_{v}M_{v^{\prime}}}} \\{= {{\psi\left( {v\;\tau^{\prime}} \right)}L_{\tau + \tau^{\prime}}M_{v + v^{\prime}}}} \\{= {{\psi\left( {v\;\tau^{\prime}} \right)}{\pi\left( {v + v^{\prime}} \right)}}} \\{= {{\Pi\left( {{\psi\left( {v\;\tau^{\prime}} \right)}\delta_{v + v^{\prime}}} \right)}.}}\end{matrix} & (3.18)\end{matrix}$

On the other hand:

$\begin{matrix}\begin{matrix}{{\delta_{v}*_{t}\delta_{v^{\prime}}} = {{\psi\left( {\beta\left( {v,v^{\prime}} \right)} \right)}\delta_{v}*\delta_{v^{\prime}}}} \\{= {{\psi\left( {v\;\tau^{\prime}} \right)}{\delta_{v + v^{\prime}}.}}}\end{matrix} & (3.19)\end{matrix}$

Consequently:Π(δ_(v)*_(t)δ_(v′))=ψ(vτ′)π(v+v′)  (3.20)

Hence we verified that: Π(δ_(v)*_(t)δ_(v′))=Π(δ_(v))∘Π(δ_(v′)).

3.4 Fundamental Channel Equation

We conclude this section with formulating a fundamental equationrelating the following structures:

-   -   1. Cross ambiguity function.    -   2. Ambiguity function    -   3. Channel transformation.    -   4. Twisted convolution.

This fundamental equation is pivotal to the two dimensional channelmodel that will be discussed in the next section. Let gϵH be a waveformof unit norm. Let hϵC(V). We denote by H the channel transformation:H=Π(h).  (3.21)

Theorem 3.3 (Fundamental channel equation). The following equationholds:A _(g,H(g)) =h* _(t) A _(g).  (3.22)

In words, the fundamental equation, (3.22), asserts that the crossambiguity function of g with H(g) is the twisted convolution of h withthe ambiguity function of g.

4. The Continuous OTFS Transceiver

In this section we describe a continuous variant of the OTFStransceiver.

4.1 Set-Up

The definition of the continuous OTFS transceiver assumes the followingdata:

-   -   1. Communication lattice. An undersampled lattice:        Λ⊂V,

where vol(Λ)=μ, for some μ≥1.

-   -   2. Generator waveform. A waveform of unit norm:        gϵH,

satisfying the orthogonality condition A_(g)(λ)=0 for every non-zeroelement λϵΛ^(x).

-   -   3. 2D filter. A window function:        WϵC(Λ).

We note that, typically, the support of the 2D filter along the delayand Doppler dimensions is bounded by the latency and bandwidthrestrictions of the communication packet respectively.

Example 4.1

A typical example of a communication lattice is the standardcommunication lattice:Λ_(T,μ) =ZμT⊕Z1/T.

A typical example of a 2D filter is:

${W\left\lbrack {{{KT}\;\mu},{L\text{/}T}} \right\rbrack} = \left\{ {\begin{matrix}1 & {{K \in \left\lbrack {0,{n - 1}} \right\rbrack},{L \in \left\lbrack {0,{m - 1}} \right\rbrack}} \\0 & {otherwise}\end{matrix},} \right.$

where m,nϵN^(≥1) and TϵR. The parameter T is called the symbol time. Thereal numbers nμT and m/T are the latency and bandwidth of thecommunication packet respectively. Note that a more sophisticated designof a spectral window will involve some level of tapering around theboundaries in the expense of spectral efficiency. Finally, in case μ=1(critical sampling) a simple example of an orthogonal waveform is:g=1_([0,T]).

4.1.1 Generalized set-up. The set-up can be slightly generalized byassuming, instead of a single orthogonal waveform g, a pair consistingof transmit waveform g_(t)ϵH and receive waveform g_(r)ϵH satisfying thefollowing cross orthogonality condition:A _(g) _(r) _(,g) _(t) (λ)=0,  (4.1)for every λϵΛ^(x). The trade-off in using a pair where g_(r)≠g_(t) isgaining more freedom in the design of the shape of each waveform inexpense of lower effective SNR at the receiver. For the sake ofsimplicity, in what follows we will consider only the case wheng_(r)=g_(t) with the understanding that all results can be easilyextended to the more general case.

4.2 Continuous OTFS Modulation Map. Let Z^(⊥) denote the torusassociated with the lattice Λ^(⊥) reciprocal to the communicationlattice. The continuous OTFS modulation map is the linear transformationM:C(Z^(|))→H, given by:M(x)=Π(W·SF _(Λ) ⁻¹(x))g,  (4.2)for every xϵ C(Z⁻). Roughly, the continuous OTFS modulation is thecomposition of the Heisenberg representation with the (inverse) discretesymplectic Fourier transform. In this regard it combines the twointrinsic structures of the delay Doppler plane. Formula (4.2) can bewritten more explicitly as:

$\begin{matrix}{{{M(x)} = {\sum\limits_{\lambda \in \Lambda}{{W(\lambda)}{X(\lambda)}{\pi(\lambda)}g}}},} & (4.3)\end{matrix}$where X=SF_(Λ) ⁻¹(x).

FIG. 27 illustrates an exemplary structure of the OTFS modulation map.Note that FIG. 27 includes an additional spreading transformation givenby convolution with a specifically designed function αϵC(Z^(⊥)). Theeffect of this convolution is to spread the energy of each informationsymbol uniformaly along the torus Z^(⊥) achieving a balanced powerprofile of the transmit waveform depending only on the total energy ofthe information vector x.

4.3 Continuous OTFS Demodulation Map

The continuous OTFS demodulation map is the linear transformationD:H→C(Z^(⊥)), given by:D(ϕ)=SF _(Λ)( W·R ^(Λ)(A _(g,ϕ))),  (4.4)for every ϕϵH. Roughly, the continuous OTFS demodulation map is thecomposition of the discrete symplectic Fourier transform with the Wignertransform. Formula (4.4) can be written more explicitly as:

$\begin{matrix}{{{{D(\varphi)}(u)} = {c \cdot {\sum\limits_{\lambda \in \Lambda}{{\psi\left( {- {\omega\left( {u,\lambda} \right)}} \right)}{\overset{\_}{W}(\lambda)}\left\langle {{{\pi(\lambda)}g},\varphi} \right\rangle}}}},} & (4.5)\end{matrix}$

for every ϕϵH and uϵZ^(⊥).

4.4 Two Dimensional Channel Model

Before describing the technical details of the two-dimensional channelmodel for the OTFS transceiver, we will provide an overview insimplified terms. Consider first that in the standard one-dimensionalphysical coordinates of time (or frequency), the wireless channel is acombination of multipath moving reflectors that induce a distortion onthe transmitted signal. This distortion arises due to superposition oftime delay and Doppler shifts. Such a general distortion appears instandard physical coordinates as a fading non-stationary inter-symbolinterference pattern. In contrast, when converted to the coordinates ofthe OTFS modulation torus, the distortion becomes a static twodimensional local ISI distortion. This is a novel and characteristicattribute of the OTFS transceiver. In what follows we provide a rigorousderivation of this characteristic. To this end, we begin by consideringthe simplest multipath channel H:H→H that is already a combination oftime delay and Doppler shift. In our terminology, this channel is givenby:H=Π(δ_(v) ₀ )=L _(τ) ₀ M _(v) ₀ ,  (4.6)for some v₀=(τ₀, v₀)ϵV. We assume, in addition, that the vector v₀satisfy ∥v₀∥=diam(Λ) where the diameter of the lattice is by definitionthe radius of its Voronoi region. Stated differently, we assume that thevector is small compared with the dimensions of the lattice. Note, thatthis assumption holds for most relevant scenarios in wirelessapplications. We proceed to derive the structure of the modulationequivalent channel. Let q:V→C be the quadratic exponential functiongiven by:q(v)=ψ(−β(v,v)),  (4.7)

for every vϵV.

Proposition 4.2 The modulation equivalent channel y=D∘H∘M(x) is aperiodic convolution y=h_(eqv)*x, where the impulse response h_(eqv)ϵC(Z^(⊥)) is given by:h _(eqv) =R _(Λ) _(⊥) (q(v ₀)δ_(v) ₀ )*SF _(Λ) |W| ²,  (4.8)That is, Equation (4.8) states that the modulation equivalent channel isa periodic convolution with a periodic blurred version of q(v₀)δ_(v) ₀where the blurring pulse is given by the symplectic Fourier transform ofthe discrete pulse |W|². This blurring results in a resolution losswhich is due to the spectral truncation imposed by the filter W. As aresult, the resolution improves as the window size increases (whatamounts to longer latency and wider bandwidth). Granting the validity ofEquation (4.8), it is straightforward to deduce the modulationequivalent of a general wireless channel:H=Π(h),  (4.9)for any function hϵC(V) where we assume that the support of h is muchsmaller than the diameter of the lattice Λ. The general two dimensionalchannel model is formulated in the following theorem.

Theorem (Two dimensional channel model). The modulation equivalentchannel y=D∘H∘M(x) is a periodic convolution y=h_(eqv)*x with theimpulse response h_(eqv) ϵC(Z^(⊥)), given by:h _(eqv) =R _(Λ) _(⊥) (q·h)*SF _(Λ) |W| ².  (4.10)Stated differently, the modulation equivalent channel is a periodicconvolution with a periodic blurred version of q·h where the blurringpulse is given by the discrete symplectic Fourier transform of thediscrete pulse |W|².

4.4.1 Derivation of the Two Dimensional Channel Model.

We proceed to derive Equation (4.8). Let xϵC(Z^(⊥)). Let ϕ_(t)ϵH denotethe transmitted signal. We have:

$\begin{matrix}\begin{matrix}{\varphi_{t} = {M(x)}} \\{{= {{\Pi\left( {W \cdot X} \right)}g}},}\end{matrix} & (4.11)\end{matrix}$

where X=SF_(Λ) ⁻¹(x). Let ϕ_(r)ϵH denote the received signal. We have:

$\begin{matrix}\begin{matrix}{\varphi_{r} = {H\left( \varphi_{t} \right)}} \\{= {{{\Pi\left( \delta_{v_{0}} \right)} \circ {\Pi\left( {W \cdot X} \right)}}g}} \\{{= {{\Pi\left( {\delta_{v_{0}}*_{t}\left( {W \cdot X} \right)} \right)}g}},}\end{matrix} & (4.12)\end{matrix}$where the third equality follows from the Heisenberg property of the mapΠ (Theorem 3.1). The demodulated vector y=D(ϕ_(r)) is given by:D(ϕ_(r))=SF _(Λ)( W·R ^(Λ)(A _(g,ϕ) _(r) )).  (4.13)

We now evaluate the right hand side of (4.13) term by term. Applying thefundamental channel equation (Theorem 3.3) we get:A _(g,ϕ) _(r) =δ_(v) ₀ *_(t)(W·X)*_(t) A _(g).  (4.14)

Considering the restriction R^(Λ)(A_(g,ϕ) _(r) ) we have the followingproposition.

Proposition. We haveR ^(Λ)(A _(g,ϕ) _(r) ),q(v ₀)R ^(Λ)(ψ_(v) ₀ )·(WX),  (4.15)

where ψ_(v) ₀ (v)=ψ(ω(v₀, v)) for every vϵV.

Combining Equations (4.13) and (4.15) we get:

$\begin{matrix}{{D\left( \varphi_{r} \right)};{{{q\left( v_{0} \right)}{{SF}_{\Lambda}\left( {\left\lfloor {{R^{\Lambda}\left( \psi_{v_{0}} \right)}{W}^{2}} \right\rfloor \cdot X} \right)}} = {\left\lfloor {{R_{\Lambda^{\bot}}\left( {{q\left( v_{0} \right)}\delta_{v_{0}}} \right)}*{{SF}_{\Lambda}\left( {W}^{2} \right)}} \right\rfloor*{x.}}}} & (4.16)\end{matrix}$

This concludes the derivation of the two dimensional channel model.

4.5 Explicit Interpretation

We conclude this section by interpreting the continuous OTFS modulationmap in terms of classical DSP operations. We use in the calculations thestandard communication lattice Λ=Λ_(T,μ) from Example 1.1. Recall thedefinition of the continuous modulation map:

$\begin{matrix}{{{M(x)} = {\sum\limits_{\lambda \in \Lambda}{{W(\lambda)}{X(\lambda)}{\pi(\lambda)}g}}},} & (4.17)\end{matrix}$for every xϵC(Z^(⊥)), where X=SF_(Λ) ⁻¹(x). Formula (4.17) can bewritten more explicitly as:

$\begin{matrix}{{{M(x)} = {\sum\limits_{K,L}{{W\left\lbrack {{K\;\mu\; T},{L\text{/}T}} \right\rbrack}{X\left\lbrack {{K\;\mu\; T},{L\text{/}T}} \right\rbrack}L_{{KT}\;\mu}{M_{L\text{/}T}(g)}}}}{\sum\limits_{K}{L_{{KT}\;\mu}{\sum\limits_{L}{{W_{K}\left\lbrack {L\text{/}T} \right\rbrack}{X_{K}\left\lbrack {L\text{/}T} \right\rbrack}{M_{L\text{/}T}(g)}}}}} = {\sum\limits_{K}{{{L_{{KT}\;\mu}\left( \phi_{K} \right)}.{where}}\text{:}}}} & (4.18) \\{\phi_{K} = {\sum\limits_{L}{{W_{K}\left\lbrack {L\text{/}T} \right\rbrack}{X_{K}\left\lbrack {L\text{/}T} \right\rbrack}{{M_{L\text{/}T}(g)}.}}}} & (4.19)\end{matrix}$

The waveform ϕ_(K) is called the Kth modulation block.

4.5.1 Frequency domain interpretation. Let G denote the Fouriertransform of g. Equation (4.19) can be interpreted as feeding theweighted sequence W_(K)X_(K) into a uniform filterbank with eachsubcarrier shaped by the filter G. See FIG. 28.

4.5.2 Time Domain Interpretation.

Let w_(K) and x_(K) denote the inverse discrete Fourier transform of thediscrete waveforms W_(K) and X_(K) respectively. Both waveforms areperiodic with period T. We have:ϕ_(K)∝(w _(K) *x _(K))·g,where * stands for periodic convolution. The waveform x_(K) can beexpressed in terms of the information vector x as follows:

x_(K)(t) ∝ ∫_(v)ψ(KT μ v)x(t, v)dv,In words, x_(K)(t) is proportional to the Kth component of the inversediscrete Fourier transform of x along the Doppler dimension.5 The Finite OTFS Transceiver

In this section we describe a finite variant of the OTFS transceiver.This variant is obtained, via uniform sampling, from the continuousvariant described previously.

5.1 Set-Up

The definition of the finite OTFS transceiver assumes the following:

-   -   1. Communication lattice. An undersampled lattice:        Λ⊂V,

where vol(Λ)=μ, for some μ≥1.

-   -   2. Communication sublattice. A sublattice:        Λ₀⊂Λ    -   3. Generator waveform. A waveform of unit norm:        gϵH,

satisfying the orthogonality condition A_(g)(λ)=0 for every λϵΛ^(x).

-   -   4. 2D filter. A window function:        WϵC(Λ).

Note that the support of the 2D filter is typically compatible with theconfiguration of the sublattice, as demonstrated in the followingexample.

Example 5.1

The standard nested pair of communication lattice and sublattice is:Λ=Λ_(T,μ) =ZμT⊕Z1/T,Λ₀=(Λ_(T,μ))_(n,m) =ZnμT⊕Zm/T,

where m,nϵN^(≥1) and TϵR is a parameter called the symbol time. The realnumbers nμT and m/T are the latency and bandwidth of the communicationpacket respectively. A typical compatible 2D filter is:

${W\left\lbrack {{{KT}\;\mu},{L\text{/}T}} \right\rbrack} = \left\{ {\begin{matrix}1 & {{K \in \left\lbrack {0,{n - 1}} \right\rbrack},{L \in \left\lbrack {0,{m - 1}} \right\rbrack}} \\0 & {otherwise}\end{matrix},} \right.$More sophisticated designs of a spectral window may involve, forexample, some level of tapering around the boundaries at the expense ofspectral efficiency. Finally, in case μ=1, a simple example oforthogonal waveform is:g=1_([0,T]).

5.2 Finite OTFS Modulation Map

Let Λ^(⊥)⊂Λ₀ ^(⊥) be the reciprocal nested pair. Let Z₀ ^(⊥)⊂Z^(⊥) bethe finite reciprocal torus. The finite OTFS modulation map is thelinear transformation M_(f):C(Z₀ ^(⊥))→H, defined by:M _(f)(x)=Π(W·SF _(Z) ₀ ⁻¹(x))g,  (5.1)

for every information vector xϵC(Z₀ ^(⊥)). Formula (5.1) can be writtenmore explicitly as:

${{M_{f}(x)} = {\sum\limits_{\lambda \in \Lambda}{{W(\lambda)}{X(\lambda)}{\pi(\lambda)}g}}},$

where X=SF_(Z) ₀ ⁻¹(x).

5.3 Finite OTFS Demodulation Map

The finite OTFS demodulation map is the linear transformationD_(f):H→C(Z₀ ^(⊥)), given by:D _(f)(ϕ)=SF _(Z) ₀ (R _(Λ) ₀ ( W·R ^(Λ) A _(g,ϕ))),  (5.2)

for every ϕϵH. Formula (5.2) can be written more explicitly as:

${{{D_{f}(\varphi)}(\mu)} = {c{\sum\limits_{\lambda \in Z_{0}}{{\psi\left( {- {\omega\left( {\mu,\lambda} \right)}} \right)}{\overset{\_}{W}(\lambda)}\left\langle {{\pi(\lambda)},g,\varphi} \right\rangle}}}},$

for every ϕϵH and λϵΛ₀ ^(⊥). Recall that the normalization coefficientc=vol(Λ).

5.4 The Finite Two Dimensional Channel Model

Let H=Π(h) be the channel transformation where hϵC(V) is assumed to havesmall support compared with the dimensions of the communication lattice.Recall the quadratic exponential:q(v)=ψ(−β(v,v)).

Theorem 5.2 (Finite 2D channel model). The finite modulation equivalentchannel y=D_(f)∘H∘M_(f)(x) is a cyclic convolution y=h_(eqv,f)*x withthe impulse response h_(eqv,f) ϵC(Z₀ ^(⊥)), given by:h _(eqv,f) =R ^(Λ) ⁰ ^(⊥) (R _(Λ) _(⊥) (q·h)*SF _(Λ) |W| ²).  (5.3)

FIG. 18 demonstrates the statement of this theorem. The bar diagram 1810represents the transmitted information vector x. The bar diagram 1820represents the received information vector y. The bar diagram 1830represents the 2D impulse response realizing the finite modulationequivalent channel. The received vector is related to the transmitvector by 2D cyclic convolution with the 2D impulse response. Finally,we see from Equation (5.3) that the finite impulse response h_(eqv,f) isthe sampling of the continuous impulse response h_(eqv) on the finitesubtorus Z₀ ^(⊥)⊂Z^(⊥).

5.5 Explicit Interpretation

We conclude this section by interpreting the finite OTFS modulation mapin terms of classical DSP operations. We use in the calculations thenested pair Λ₀⊂Λ from example 5.1. Recall the definition of the finitemodulation map:

$\begin{matrix}{{{M_{f}(x)} = {\sum\limits_{\lambda \in \Lambda}{{W(\lambda)}{X(\lambda)}{\pi(\lambda)}g}}},} & (5.4)\end{matrix}$

for every xϵC(Z₀ ^(ϵ)), where X=SF_(Z) ₀ ⁻¹(x). Formula (5.4) can bewritten more explicitly as:

$\begin{matrix}{{{M_{f}(x)} = {\sum\limits_{K,L}{{W\left\lbrack {{K\;\mu\; T},{L\text{/}T}} \right\rbrack}{X\left\lbrack {{K\;\mu\; T},{L\text{/}T}} \right\rbrack}L_{{KT}\;\mu}{M_{L\text{/}T}(g)}}}}{\sum\limits_{K}{L_{{KT}\;\mu}{\sum\limits_{L}{{W_{K}\left\lbrack {L\text{/}T} \right\rbrack}{X_{K}\left\lbrack {L\text{/}T} \right\rbrack}{M_{L\text{/}T}(g)}}}}} = {\sum\limits_{K}{{{L_{{KT}\;\mu}\left( \phi_{K} \right)}.{where}}\text{:}}}} & (5.5) \\{\phi_{K} = {\sum\limits_{L}{{W_{K}\left\lbrack {L\text{/}T} \right\rbrack}{X_{K}\left\lbrack {L\text{/}T} \right\rbrack}{M_{L\text{/}T}(g)}}}} & (5.6)\end{matrix}$

The waveform ϕ_(K) is called the Kth modulation block.

5.5.1 Frequency Domain Interpretation.

Let G denote the Fourier transform of g. Equation (5.6) can beinterpreted as feeding the sequence W_(K)X_(K) into a uniform filterbankwith each subcarrier shaped by the filter G.

5.5.2 Time Domain Interpretation.

Let w_(K) and x_(K) denote the inverse discrete Fourier transform of thediscrete waveforms W_(K) and X_(K) respectively. Both waveforms areperiodic with period T. We have:

${\phi_{K} \propto {\left( {w_{K}*{\sum\limits_{k}{{x_{K}\left\lbrack {{kT}\text{/}m} \right\rbrack}\delta_{{kT}\text{/}m}}}} \right) \cdot g}},$where * stands for periodic convolution. The waveform x_(K) can beexpressed in terms of the information vector x as:

${{x_{K}\left( {{kT}\text{/}m} \right)} \propto {\sum\limits_{l = 0}^{n - 1}\;{{\psi({Kl})}{x\left\lbrack {\frac{kT}{m},\frac{l}{{nT}\;\mu}} \right\rbrack}}}},$In words, x_(K) is proportional to the inverse finite Fourier transformof x along the Doppler dimension.

While various embodiments have been described above, it should beunderstood that they have been presented by way of example only, and notlimitation. They are not intended to be exhaustive or to limit theclaims to the precise forms disclosed. Indeed, many modifications andvariations are possible in view of the above teachings. The embodimentswere chosen and described in order to best explain the principles of thedescribed systems and methods and their practical applications, theythereby enable others skilled in the art to best utilize the describedsystems and methods and various embodiments with various modificationsas are suited to the particular use contemplated.

Where methods described above indicate certain events occurring incertain order, the ordering of certain events may be modified.Additionally, certain of the events may be performed concurrently in aparallel process when possible, as well as performed sequentially asdescribed above. Although various modules in the different devices areshown to be located in the processors of the device, they can also belocated/stored in the memory of the device (e.g., software modules) andcan be accessed and executed by the processors. Accordingly, thespecification is intended to embrace all such modifications andvariations of the disclosed embodiments that fall within the spirit andscope of the appended claims.

The foregoing description, for purposes of explanation, used specificnomenclature to provide a thorough understanding of the claimed systemsand methods. However, it will be apparent to one skilled in the art thatspecific details are not required in order to practice the systems andmethods described herein. Thus, the foregoing descriptions of specificembodiments of the described systems and methods are presented forpurposes of illustration and description. They are not intended to beexhaustive or to limit the claims to the precise forms disclosed;obviously, many modifications and variations are possible in view of theabove teachings. The embodiments were chosen and described in order tobest explain the principles of the described systems and methods andtheir practical applications, they thereby enable others skilled in theart to best utilize the described systems and methods and variousembodiments with various modifications as are suited to the particularuse contemplated. It is intended that the following claims and theirequivalents define the scope of the systems and methods describedherein.

The various methods or processes outlined herein may be coded assoftware that is executable on one or more processors that employ anyone of a variety of operating systems or platforms. Additionally, suchsoftware may be written using any of a number of suitable programminglanguages and/or programming or scripting tools, and also may becompiled as executable machine language code or intermediate code thatis executed on a framework or virtual machine.

Examples of computer code include, but are not limited to, micro-code ormicro-instructions, machine instructions, such as produced by acompiler, code used to produce a web service, and files containinghigher-level instructions that are executed by a computer using aninterpreter. For example, embodiments may be implemented usingimperative programming languages (e.g., C, Fortran, etc.), functionalprogramming languages (Haskell, Erlang, etc.), logical programminglanguages (e.g., Prolog), object-oriented programming languages (e.g.,Java, C++, etc.) or other suitable programming languages and/ordevelopment tools. Additional examples of computer code include, but arenot limited to, control signals, encrypted code, and compressed code.

In this respect, various inventive concepts may be embodied as acomputer readable storage medium (or multiple computer readable storagemedia) (e.g., a computer memory, one or more floppy discs, compactdiscs, optical discs, magnetic tapes, flash memories, circuitconfigurations in Field Programmable Gate Arrays or other semiconductordevices, or other non-transitory medium or tangible computer storagemedium) encoded with one or more programs that, when executed on one ormore computers or other processors, perform methods that implement thevarious embodiments of the invention discussed above. The computerreadable medium or media can be transportable, such that the program orprograms stored thereon can be loaded into one or more differentcomputers or other processors to implement various aspects of thepresent invention as discussed above.

The terms “program” or “software” are used herein in a generic sense torefer to any type of computer code or set of computer-executableinstructions that can be employed to program a computer or otherprocessor to implement various aspects of embodiments as discussedabove. Additionally, it should be appreciated that according to oneaspect, one or more computer programs that when executed perform methodsof the present invention need not reside on a single computer orprocessor, but may be distributed in a modular fashion amongst a numberof different computers or processors to implement various aspects of thepresent invention.

Computer-executable instructions may be in many forms, such as programmodules, executed by one or more computers or other devices. Generally,program modules include routines, programs, objects, components, datastructures, etc. that perform particular tasks or implement particularabstract data types. Typically the functionality of the program modulesmay be combined or distributed as desired in various embodiments.

Also, data structures may be stored in computer-readable media in anysuitable form. For simplicity of illustration, data structures may beshown to have fields that are related through location in the datastructure. Such relationships may likewise be achieved by assigningstorage for the fields with locations in a computer-readable medium thatconvey relationship between the fields. However, any suitable mechanismmay be used to establish a relationship between information in fields ofa data structure, including through the use of pointers, tags or othermechanisms that establish relationship between data elements.

Also, various inventive concepts may be embodied as one or more methods,of which an example has been provided. The acts performed as part of themethod may be ordered in any suitable way. Accordingly, embodiments maybe constructed in which acts are performed in an order different thanillustrated, which may include performing some acts simultaneously, eventhough shown as sequential acts in illustrative embodiments.

All definitions, as defined and used herein, should be understood tocontrol over dictionary definitions, definitions in documentsincorporated by reference, and/or ordinary meanings of the definedterms.

The indefinite articles “a” and “an,” as used herein in thespecification and in the claims, unless clearly indicated to thecontrary, should be understood to mean “at least one.”

The phrase “and/or,” as used herein in the specification and in theclaims, should be understood to mean “either or both” of the elements soconjoined, i.e., elements that are conjunctively present in some casesand disjunctively present in other cases. Multiple elements listed with“and/or” should be construed in the same fashion, i.e., “one or more” ofthe elements so conjoined. Other elements may optionally be presentother than the elements specifically identified by the “and/or” clause,whether related or unrelated to those elements specifically identified.Thus, as a non-limiting example, a reference to “A and/or B”, when usedin conjunction with open-ended language such as “comprising” can refer,in one embodiment, to A only (optionally including elements other thanB); in another embodiment, to B only (optionally including elementsother than A); in yet another embodiment, to both A and B (optionallyincluding other elements); etc.

As used herein in the specification and in the claims, “or” should beunderstood to have the same meaning as “and/or” as defined above. Forexample, when separating items in a list, “or” or “and/or” shall beinterpreted as being inclusive, i.e., the inclusion of at least one, butalso including more than one, of a number or list of elements, and,optionally, additional unlisted items. Only terms clearly indicated tothe contrary, such as “only one of” or “exactly one of,” or, when usedin the claims, “consisting of,” will refer to the inclusion of exactlyone element of a number or list of elements. In general, the term “or”as used herein shall only be interpreted as indicating exclusivealternatives (i.e. “one or the other but not both”) when preceded byterms of exclusivity, such as “either,” “one of,” “only one of,” or“exactly one of.” “Consisting essentially of,” when used in the claims,shall have its ordinary meaning as used in the field of patent law.

As used herein in the specification and in the claims, the phrase “atleast one,” in reference to a list of one or more elements, should beunderstood to mean at least one element selected from any one or more ofthe elements in the list of elements, but not necessarily including atleast one of each and every element specifically listed within the listof elements and not excluding any combinations of elements in the listof elements. This definition also allows that elements may optionally bepresent other than the elements specifically identified within the listof elements to which the phrase “at least one” refers, whether relatedor unrelated to those elements specifically identified. Thus, as anon-limiting example, “at least one of A and B” (or, equivalently, “atleast one of A or B,” or, equivalently “at least one of A and/or B”) canrefer, in one embodiment, to at least one, optionally including morethan one, A, with no B present (and optionally including elements otherthan B); in another embodiment, to at least one, optionally includingmore than one, B, with no A present (and optionally including elementsother than A); in yet another embodiment, to at least one, optionallyincluding more than one, A, and at least one, optionally including morethan one, B (and optionally including other elements); etc.

In the claims, as well as in the specification above, all transitionalphrases such as “comprising,” “including,” “carrying,” “having,”“containing,” “involving,” “holding,” “composed of,” and the like are tobe understood to be open-ended, i.e., to mean including but not limitedto. Only the transitional phrases “consisting of” and “consistingessentially of” shall be closed or semi-closed transitional phrases,respectively, as set forth in the United States Patent Office Manual ofPatent Examining Procedures, Section 2111.03.

What is claimed is:
 1. A method for waveform transmission, the methodcomprising: receiving a plurality of information symbols: creating amatrix of modulation parameters by using each of the pluralityinformation symbols to modulate one of a plurality of two-dimensionalbasis functions on a time-frequency plane wherein each of the pluralityof two-dimensional basis functions is uniquely associated with one ofthe plurality of information symbols; generating a transmit waveformcomprised of a plurality of pulse waveforms, each of the plurality ofpulse waveforms corresponding to a combination of one of the modulationparameters and one of a plurality of time-translated andfrequency-modulated versions of a fundamental transmit pulse; andtransmitting the transmit waveform over a communication channel.
 2. Themethod of claim 1 wherein each of the plurality of time-translated andfrequency-modulated versions of the fundamental transmit pulse isassociated with a time translation by one of N multiples of T and afrequency modulation by one of the M multiples of Δƒ, where N and M areconstants, N×M is equivalent to a number of the modulation parameters, Tis a time sampling period, and Δƒ is a frequency sampling period.
 3. Themethod of claim 2 wherein the fundamental transmit pulse has theproperty of being orthogonal to translations by a multiple of time T andmodulation by a multiple of Δƒ.
 4. The method of claim 2 wherein thetransmit waveform is of a total duration of NT seconds and a totalbandwidth of MΔƒ Hz.
 5. The method of claim 1 further includinggenerating the plurality of pulse waveforms by modulating a set of OFDMsubcarriers.
 6. The method of claim 2 further including arranging theplurality of information symbols on a two-dimensional grid having afirst dimension of N elements and a second dimension of M elements.
 7. Acommunication device, comprising: a wireless transmitter; a processor;and a memory including program code executable by the processor, theprogram code including: code for causing the processor to receive aplurality of information symbols; code for causing the processor tocreate a matrix of modulation parameters by using each of the pluralityinformation symbols to modulate one of a plurality of two-dimensionalbasis functions on a time-frequency plane wherein each of the pluralityof two-dimensional basis functions is uniquely associated with one ofthe plurality of information symbols; code for causing the processor togenerate a transmit waveform comprised of a plurality of pulsewaveforms, each of the plurality of pulse waveforms corresponding to acombination of one of the modulation parameters and one of a pluralityof time-translated and frequency-modulated versions of a fundamentaltransmit pulse; and code for causing the processor to provide thetransmit waveform to the wireless transmitter for transmission by thewireless transmitter.
 8. The communication device of claim 7 whereineach of the plurality of time-translated and frequency-modulatedversions of the fundamental transmit pulse is associated with a timetranslation by one of N multiples of T and a frequency modulation by oneof the M multiples of Δƒ, where N and M are constants, N×M is equivalentto a number of the modulation parameters, T is a time sampling period,and Δƒ is a frequency sampling period.
 9. The communication device ofclaim 7 wherein the fundamental transmit pulse has the property of beingorthogonal to translations by a multiple of time T and modulation by amultiple of Δƒ.
 10. The communication device of claim 8 wherein thetransmit waveform is of a total duration of NT seconds and a totalbandwidth of MΔƒ Hz.
 11. The communication device of claim 7, whereinthe program code further includes code for generating the plurality ofpulse waveforms by modulating a set of OFDM subcarriers.
 12. Thecommunication device of claim 7, wherein the program code furtherincludes code for arranging the plurality of information symbols on atwo-dimensional grid having a first dimension of N elements and a seconddimension of M elements.
 13. A method, comprising: receiving, at acommunication receiver, one or more modulated waveforms; matchedfiltering samples of the one or more modulated waveforms with respect toa receive pulse to produce estimated time-frequency modulation symbolswherein each of the estimated time-frequency modulation symbolscorresponds to modulation of one of a plurality of orthogonaltwo-dimensional basis functions by one of a plurality of informationsymbols; and projecting the estimates of the time-frequency modulationsymbols on the plurality of orthogonal two-dimensional basis functionsin order to obtain estimates of the plurality of information.
 14. Themethod of claim 13 further including performing windowing andperiodization operations with respect to the estimated time-frequencymodulation symbols.
 15. The method of claim 13 wherein the projectingincludes performing a symplectic Fourier transform operation withrespect to a periodic sequence comprised of the estimates of thetime-frequency modulation symbols.
 16. A communication device,comprising: a wireless receiver configured to receive one or moremodulated waveforms; a processor; and a memory including program codeexecutable by the processor, the program code including: code forcausing the processor to receive, from the wireless receiver, samples ofone or more modulated waveforms, code for causing the processor tomatched filter samples of the one or more modulated waveforms withrespect to a receive pulse to produce estimated time-frequencymodulation symbols wherein each of the estimated time-frequencymodulation symbols corresponds to modulation of one of a plurality oforthogonal two-dimensional basis functions by one of a plurality ofinformation symbols; and code for causing the processor to project theestimated time-frequency modulation symbols on the plurality oforthogonal two-dimensional basis functions in order to obtain estimatesof the plurality of information symbols.
 17. The communication device ofclaim 16 wherein the program code further includes code for performingwindowing and periodization operations with respect to the estimatedtime-frequency modulation symbols.
 18. The communication device of claim16 wherein the code for causing the processor to project furtherincludes code for causing the processor to perform a symplectic Fouriertransform operation with respect to a periodic sequence comprised of theestimated time-frequency modulation symbols.
 19. The method of claim 13wherein the projecting is in accordance with equation:${\hat{x}\left\lbrack {l,k} \right\rbrack} = {\sum\limits_{m = 0}^{M - 1}\;{\sum\limits_{n = 0}^{N - 1}\;{{\hat{X}\left( {n,m} \right)}{b_{k,l}^{*}\left( {n,m} \right)}}}}$where {circumflex over (x)}[l, k] represents the estimates of theplurality of information symbols, where {circumflex over (X)}(n, m)represent the estimated time-frequency modulation symbols, and where${b_{k,l}^{*}\left( {n,m} \right)} = e^{{- j}\; 2{\pi{({\frac{lm}{L} - \frac{kn}{K}})}}}$corresponds to the plurality of orthogonal two-dimensional basisfunctions.
 20. The communication device of claim 16 wherein the code forcausing the processor to project is in accordance with equation:${\hat{x}\left\lbrack {l,k} \right\rbrack} = {\sum\limits_{m = 0}^{M - 1}\;{\sum\limits_{n = 0}^{N - 1}\;{{\hat{X}\left( {n,m} \right)}{b_{k,l}^{*}\left( {n,m} \right)}}}}$where {circumflex over (x)}[l, k] represents the estimates of theplurality of information symbols, where {circumflex over (X)}(n, m)represent the estimated time-frequency modulation symbols, and where${b_{k,l}^{*}\left( {n,m} \right)} = e^{{- j}\; 2{\pi{({\frac{lm}{L} - \frac{kn}{K}})}}}$corresponds to the plurality of orthogonal two-dimensional basisfunctions.
 21. The method of claim 1 wherein the plurality of modulationsymbols are created in accordance with the equation:${X\left\lbrack {n,m} \right\rbrack} = {\frac{1}{MN}{W_{tr}\left\lbrack {n,m} \right\rbrack}{\sum\limits_{k = 0}^{N - 1}\;{\sum\limits_{l = 0}^{M - 1}\;{{x\left\lbrack {l,k} \right\rbrack}{b_{k,l}\left\lbrack {n,m} \right\rbrack}}}}}$${b_{k,l}\left\lbrack {n,m} \right\rbrack} = e^{j\; 2{\pi{({\frac{ml}{M} - \frac{nk}{N}})}}}$where X[n, m], n=0, . . . , N−1, m=0, . . . , M−1 represent theplurality of modulation symbols, x[k, l] represents the plurality ofinformation symbols, W_(tr) [n, m] represents a transmit windowingfunction, and b_(k,l)[n, m] represents the plurality of two-dimensionalbasis functions.